Exponential upper-bound conjecture for the strong Tic-Tac-Toe threshold

Let $q$ be a prime power. In the strong $(m,n)_q$-Tic-Tac-Toe game on the affine space $4\mathbb{F}_q^m4$, let $T(n,q)$ denote the smallest dimension $m$ for which the first player has a winning strategy. Exponential threshold conjecture. For every prime power $q$, there exists a constant $C_q>0$ such that

$$T(n,q)\le C_q q^n$$

for every $n\ge 1$. The conjecture is motivated by the known exponential upper bound for $q=2$ and the corresponding Erdős–Selfridge lower bound; establishing such an upper bound for every prime power remains open.