Interacting CFT in $d>6$ There's an expectation that there aren't interacting CFT in $d>6$.
As I understand, main reason for this is due to the scaling dimension of ordinary scalar fields and Dirac fields. This lead to absence of relevant operators and then to existence only of Gaussian fixed points with free CFT.
But on other side, CFT data must satisfy some bootstrap equations, and as I understand, there's no any rigorous statements about absence of solutions of such equations.
Why is it hard to construct such theories? Which fundamental principles may prohibit existence of such theories?
 A: I would say that the claim that there are no non-trivial CFTs in $d>6$ is just a speculation for which there isn't much evidence.
The belief is that above six spacetime dimensions, the only unitary CFTs are simply free theories and that all non-trivial fixed points can be described by mean field theory. In addition to what you said, that there are fewer relevant operators as we increase $d$, part of the expectation comes from the fact that there are no superconformal CFTs in $d>6$ (as the superconformal algebra cannot be defined). But apart from these two facts, I’d say that there’s really no evidence for the absence of interacting CFTs in $d>6$. From a bootstrap perspective, nothing strange happens (that I’m aware of) for the conformal blocks in these dimensions, and as was linked to in a comment, the large $d$ asymptotics of conformal blocks work pretty well. I think there are other rough arguments, but definitely no strong evidence.
A: Actually...
There is some evidence of CFTs in $d>6$. In [1] they construct a solution in AdS$_8$ implies the existence of a CFT in $d=7$.
This is not a definitive answer because there are still some issues about the solution. One has to prove full nonperturbative stability and also there is a region in spacetime where the coupling becomes big and one has to motivate that the supergravity effective theory is still valid. So it's not a proof, but something to keep in mind nontheless.
But to answer your question
The expectation came from the fact that are no Lagrangians with relevant couplings in $d>6$. If you just take a scalar model for instance, the vertex $\varphi^3$ has dimension $\frac32(d-2)$ which is bigger than $d$ if $d>6$. This automatically means that you can't play the usual game of writing down a Lagrangian and tweaking the parameters so that the $\beta$ function vanishes (that is, if you want more than just free theories).
So the only CFTs in $d>6$ are

*

*Free theories: boring

*Non Lagrangian theories: difficult to find, so people hoped they wouldn't exist.

Another speculation for the lore is that one can mathematically prove that there are no superconformal field theories in $d>6$. So I guess it felt natural to think that this pattern would carry over to non-supersymmetric theories as well. (I don't feel this would be a strong motivation, but I wanted to mention it anyway.)

[1] AdS$_8$ Solutions in Type II Supergravity, Clay Cordova, G. Bruno De Luca, Alessandro Tomasiello, 1811.06987
