There is some evidence of CFTs in $d>6$. In  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.)
 AdS$_8$ Solutions in Type II Supergravity, Clay Cordova, G. Bruno De Luca, Alessandro Tomasiello, 1811.06987