I am interested in determining the the number of indecomposable modules of finite groups over finite fields of a fixed dimension. Specifically, I have the following conjecture:

Conjecture. Suppose we have a finite group $G$ of order $n$, and $F_p$ the field of size $p$, $p$ a prime. Let $s(k)$ be the number of non-isomorphic indecomposable modules of dimension $k$ and over $F_p$, for the group algebra $F_pG$. Then $s(k)$ is bounded by a polynomial in $n$ and $p^k$.

I have no idea whether this conjecture is true or not -- I suspect it not correct in general, but could not prove it. The second Brauer-Thrall conjecture (now theorem) is somehow related but does not address the case of finite fields. I would be very grateful if anyone could give some hints of references for this. Thank you!

Best, Jimmy