Vanishing of first co-homology with coefficients modular representations of small dimension

Question

Is the following true:

For any $n$ there exists $p_0$ s.t. for any finite group $G$ of Lie type of rank $\leq n$ and characteristic $p\geq p_0$ and any (irreducible) $\mathbb F_p$ representation $V$ of $G$ of dimension $\leq n$  we have  $$H^1(G,V)=0$$

Answer 1

I think the answer to your question is yes for example with $p_0 = n+3$.

See the following paper:

"Small Representations Are Completely Reducible", Robert M. Guralnick, J. Algebra (1999).

Theorem A in this paper says the following:

Let $k$ be a field of positive characteristic $p$. Let $G$ be a finite group containing no nontrivial normal $p$-subgroup. Let $V$ be a $kG$-module such that $G$ acts faithfully on $V$.

(a) If $\dim V \leq p − 2$, then $V$ is completely reducible;

(b) If $\dim V \leq p − 3$, then $H^1(G, V) = 0$.