What are the strongest objections to be made against decoherence as an explanation of "collapse?" When we measure an observable A of a quantum system, we get an eigenvalue of A. Without worrying about connotations of Copenhagen vs. MWI, etc., let's just call this "collapse."
Question: Among people who are not convinced that decoherence gives a complete answer to this problem, what are the strongest objections that have been raised against its acceptance?
A somewhat cryptic take on this seems to be given in this 2013 paper by Bubb, who says on p. 20,

The argument here is not that decoherence provides a dynamical explanation of how an indefinite quantity becomes definite in a measurement process—Bell [3] has aptly criticized this argument as a ‘for all practical purposes’ (FAPP) solution to the measurement problem.  Rather, the claim is that we can take the decoherence pointer as definite by stipulation , and that decoherence then guarantees the objectivity of the macroworld,  which  resolves  the  measurement  problem  without  resorting  to  Copenhagen or neo-Copenhagen instrumentalism.

The reference is to a 1974 essay by Bell, "On wave-packet reduction in the Coleman-Hepp model," which I guess dates to long before people started talking about decoherence. The Bell paper can be found online (presumably illegally). It talks about a certain toy model, not about decoherence. The point he seems to be making is that it matters that the "collapse" described by this toy model only happens in the limit $t\rightarrow\infty$. If I'm understanding correctly what Bubb is saying, then I guess the idea would be that in decoherence, the off-diagonal elements of the density matrix fall off exponentially, but they never actually hit zero. If Bubb is saying that he considers this to be the strongest objection remaining to decoherence as an explanation of "collapse," then it would seem to me to be an extremely weak objection.
Please limit answers to the specific question asked above. I don't want to open up a can of worms about other questions such as whether some particular interpretation of quantum mechanics can/should/does/doesn't "explain" the Born rule. I'm also not interested in having decoherence explained to me. I think I already understand it at the level of Joos and Zeh, The emergence of classical properties through interaction with the environment, Z Phys B 59 (1985) 223, which should be sufficient for the present discussion. I'm also not interested in hearing people give yes/no answers as to whether decoherence does explain collapse, because obviously that will never lead to a conclusion that everyone accepts. I'm just asking for an outline of what reasons people do offer for objecting to this as a proposed mode of explanation. Also, I'm interested in whether my interpretation of Bubb's remark is correct.
 A: The simplest way to phrase the main objection is that decoherence doesn't even try to solve anything beyond "for all practical purposes." It is a way of describing entanglement with the environment that is irreversible "in practice," but what may be impractical today may well be practical tomorrow (or for a more advanced civilization even today). As indicated elsewhere, the evolution is still understood to be unitary in principle.
Roger Penrose (2004), The Road to Reality, pp. 802-803:

...the environmental-decoherence viewpoint ... maintains that state
  vector reduction [the R process] can be understood as coming about
  because the environmental system under consideration becomes
  inextricably entangled with its environment.[...] We think of the
  environment as extremely complicated and essentially 'random' [...]
  Under normal circumstances, one must regard the density matrix as some
  kind of approximation to the whole quantum truth. For there is no
  general principle providing an absolute bar to extracting information
  from the environment.[...] Accordingly, such descriptions are referred
  to as FAPP [For All Practical Purposes]
Maybe a future technology could provide means whereby quantum phase
  relations can be monitored in detail, under circumstances where
  present-day technology would simply ‘give up’. It would seem that the
  resort to a density-matrix description is a technology-dependent
  prescription! With better technology, the state-vector description
  could be maintained for longer, and the resort to a density matrix put
  off until things get really hopelessly messy!

Addendum
There are also attempts at demonstrating decoherence that is irreversible in principle. See related question: Can the Montevideo interpretation of quantum mechanics do what it claims?
A: I think most arguments in the literature can be boiled down to the point that decoherence does in no way touch the linearity of the Schrödinger equation, and thus cannot make an "or" from an "and". 
This is complicated in the literature by very technical discussions, which I would like to avoid.
Let me explain the basic point in more details. A widely referenced statement of the measurement problem is given in the paper "Three measurement problems" by Tim Maudlin, section 1 (first two pages, whole paper). He considers a spin measurement apparatus, which has the property that if you put in an electron in spin+ eigenstate, a pointer points to the left, and if the electron comes in with a spin- eigenstate, a pointer points to the right. Now it follows simply from the linearity of the Schrödinger evolution that the initial state
$$\rm\frac{1}{\sqrt{2}} \left(\left|up\right>_e + \left|down\right>_e\right) \otimes \left|ready\right>_d $$
evolves to the final state, after measuring,
$$\rm\frac{1}{\sqrt{2}} \left(\left|up\right>_e \otimes \left|LEFT\right>_d + \left|down\right>_e \otimes \left|RIGHT\right>_d\right).   $$
This looks like a pointer pointing to the left and to the right, which cannot be true and is not what we see in labs or around us. We somehow have to make an "or" out of this "and to get the correct facts.
Among people who believe that this is a problem and that decoherence doesn't solve it, the prevailing point of view seems to be the one given in a letter by Adler, which also references other literature. To solve this problem, which is in fact the so-called measurement problem, there are several options now:


*

*change the Schrödinger equation (done in so-called collapse models)

*add additional local beables (in Bell's words)/ things other than the wave function to the theory, as done in de-Broglie-Bohm theory

*postulate that measurements actually don't have unique outcomes and we somehow only perceive single outcomes when we do experiments (the many-worlds-path). 


For many practical purposes, the collapse postulate of the Copenhagen interpretation works fine. Now if you look for an explanation of the collapse, one of the three options above would serve as one. 
Decoherence can only explain why the collapse works well: Because the wave function parts that are disregarded won't interfere again with the ones we keep, the collapse is fine. So decoherence explains why you can, in many worlds, separate the worlds and say they do not come in contact again. In de-Broglie-Bohm, it explains why you can effectively collapse the wave function after a measurement. But the collapse itself is not explained, there is still the sum in the above formula, still an "and", and how to get a definite result from it can only go along one of the three lines I sketched.
A: Decoherence turns pure states into mixed states that follow classical probabilities. It does not cover the projection itself that turns the mixed states into individual (pure) eigenstates
