Why do string theorists seem to ignore cosmology? Related and sort of a follow-up question to: If string theory is inconsistent with observations, why hasn't it been rejected yet?
From the answer to that question, string theorists are aware the theory is inconsistent with cosmological observations, but they are not rejecting the theory, because 1) the conjectures aren't proven and 2) even if they are proven, it's possible to explain the cosmological observations some other way. Hence "if a metastable de Sitter space lasting for cosmological durations really is impossible in string theory, then dark energy needs to be explained in some other way, e.g. via quintessence."
On the other hand, here's a swampland conjecture that outright rejects inflation (first paragraph on page 4 of link):

In any event, once we [equate the slow-roll conditions $\epsilon, \eta < 1$ to inflation], it follows that to rule out inflation we do not necessarily need to rule out $\epsilon \ll 1$: we
just need to break one of the two slow roll conditions, and have either $\epsilon \ll 1$ or $|\eta| \ll 1$ violated at $\mathcal{O}(1)$. This is the main observation we make in this note, we will present various pieces of evidence in string theory, for this refinement. The refined de Sitter swampland criterion then is that at least one of $\epsilon$ or $|\eta|$ must be bigger than $\mathcal{O}(1)$ in string theory.

So the authors have created a "refined de Sitter swampland criterion" that was explicitly constructed to reject inflation. Huh? This doesn't sound like "I have a nice, well-motivated theory that works well with particle physics but is inconsistent with cosmology", it's as though the authors have created a swampland conjecture in order to make string theory inconsistent with cosmology. Furthermore, I would have expected such an article to first argue that the observational evidence for inflation is incorrect before presenting the alternative theory, yet the authors don't deal with the observational evidence at all. I don't get it. It looks like nonsense of the "not even science" kind, except that article has, as of time of writing, 139 citations.
Do string theorists not care about cosmology? Why do they seem to treat cosmology with contempt?
 A: I'm not a cosmologist nor a very good string theorist, but since I'm interested in this subject myself I'll have a go at answering this question. Feel free to correct me if there's something wrong.
First, remember why some people might hate having to throw string theory away. These points might seem obvious, but they are often skipped over in debates like these.

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*Most importantly, string theory unifies quantum mechanics with gravity. For mainstream QFT, its failure to include gravity obviously indicates the need for a deeper theory. It is sometimes said that string theory is dangerously close to being unfalsifiable, but on the other hand, the Standard Model is in some sense falsified, since it does not contain gravity. Currently, string theory seems to be the most interesting potential quantum gravity theory by far.

*String theory might explain the precise structure of the Standard Model: where do the precise gauge group, the particle masses and charges and the number of generations come from?

*String theory could possibly resolve several other problems, like the Higgs mass hierarchy problem.

*Lastly, string theory is well-known to be a fruitful research area that yields many surprising and interesting results connecting to both physics and pure mathematics.

So whenever you meet an enthousiastic string theorist, her feeling would be that string theory is something that we should at least take very seriously, even if we don't know yet how and if it fits onto our current experimental knowledge about our Universe.
Meanwhile, on the cosmology side, there is slightly more wiggle room than in particle physics, especially if one allows for models that naively look unnatural. For starters, we don't really know what dark matter is, and we don't officially know where dark energy comes from. The precise details of inflation, too, are still not settled, as far as I know.
The recent de Sitter swampland conjectures point out a potential conflict between cosmology and string theory. For string theorists, there are two ways out, a boring one and an exciting one.

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*The boring way out is that some swampland conjectures might simply be wrong. This clearly remains a possibility. On the other hand, while the conjectures have not rigorously been shown to be watertight, they do seem to tell us what string theory does and does not like, since typically these conjectures are supported by good evidence (meaning string-theoretic scenarios that support the conjectures). Of course, we should not rule out the possibility that one, many or all of these conjectures admit loopholes. But if we are to take string theory seriously as a potential theory of nature, we should listen to what it has to say.

*Another possibility is that string theory is trying to tell us something really nontrivial about cosmology. Perhaps rather strange cosmological models turn out to be very natural in string theory. Perhaps the universe really isn't de Sitter, but instead there is quintessence after all. As I understand it, typically what's in conflict with string theory are the cosmological models, not the observations. String theory might give us some new way to do cosmology, and with some luck it might even be testable.

Some surprising type of cosmology predicted by string theory could even count as a testable prediction that may be compared to future observations. So looking at the way string theory 'does' cosmology is definitely relevant to string theorists.
Granted, the paper you cited looks like string theorists attempting to bash inflation. However, the authors are simply claiming there appears to be a conspiracy within string theory against inflation, just like there appears to be a conspiracy within string theory against de Sitter spacetime. Perhaps string theory wants to replace inflation with something even better.
Finally, all of this of course still leaves open the possibility that string theory is incorrect. Cosmology might be a promising field in which to demonstrate this, more so than particle physics, to which string theory is infamously accomodating. If string theory really conspires against inflation, we should not ignore that; if, then, inflation nevertheless turns out to be correct, so much the worse for string theory.
A: I probably should not comment and will probably get downvoted by actual string theorists (I'm a former theoretical cosmologist/gravitational-waver who has talked to a lot of string theorists and listened to a lot of talks) but I can't resist...
String theorists (and, honestly, many theoretical physicists) would love nothing better than to be able to explain cosmology within string theory. A model which explained why we have four large spacetime dimensions, produced either a mechanism to solve the cosmological constant problem or at least a fully calculable first-principles quantum gravity model that implements the anthropic principle, and resolved the Big Bang singularity would be awesome. Any theorist in cosmology or string theory would love to write that paper.
Unfortunately, de Sitter space is hard to deal with in string theory. There are elaborate stories (eg, KKLT) that can support a positive cosmological constant, but it is a lot of work and only parts of it can be rigorously calculated. It's also not known how to handle time-dependent backgrounds in perturbative string theory (which you need for cosmology), and no one really knows what non-perturbative string theory is for spacetimes that are not asymptotically AdS. You can take limits where string theory becomes super gravity and develop field theory models within super gravity (or a super-gravity-adjacent theory) and call it "string theory inspired", but this purely low energy picture is kind of boring from a string theory perspective.
In terms of the swampland: as a string theorist said in a talk I attended, the swampland is an area where there are many provable but not interesting statements, and many interesting but not provable statements, but very few interesting and provable statements. As I understand (but I am not an expert), the conjectures that rule out normal inflationary models are more speculative. So I'm not sure the swampland claims are as clear cut as they sound in the headlines. With that in mind, I think it is good that people do not take the easy way out and try to handwave away the fact that the swampland conjecture is inconsistent with standard inflation. What I find less appealing is that some will use the conjecture to take a stand that inflation is disfavored, without a firm proof of the conjecture and without giving some alternative explanation of the observations inflation can explain.
Having said all of that, there are definitely exceptions: Eva Silverstein, for example, has done a lot of very cool work developing novel cosmological models motivated by string theory, and Nima Arkhani-Hamed has applied the amplitude methods he is working on to cosmological correlators. So it isn't fair to say as a blanket statement that string theorists ignore cosmology, even though I think it is fair that not a lot of progress has been made in explaining mainstream cosmology from first principles in string theory and there are some key roadblocks to doing so.
So... tl;dr... I don't think that string theorists ignore cosmology because they are uninterested. I think they ignore it because it is hard: no one knows how to make progress in the time-dependent backgrounds with inconvenient asymptotics relevant for cosmology. However, this is kind of a loaded question, and there are string theorists who do interesting things in cosmology (even if it is not the mainstream trend of applying AdS/CFT to condensed matter).
A: The fact that string theorists try to construct de Sitter solutions shows that they do not ignore cosmology.
I only wonder why such an effort is made for this, because standard cosmology also works in Anti de Sitter spaces.
See for example:
Brane world cosmology
Standard Cosmology on the Anti-de Sitter boundary
