Alice's winning conjecture for triangular grid graphs of order $2^m-3$

Let $T_n$ be the triangular grid graph with starting vertex $v^0_0$, and consider the feedback game on $T_n$ in which Alice and Bob play according to the game's rules. Alice's winning conjecture. If

$$n=2^m-3$$

with $m\geq 2$, then Alice wins the game on $T_n$. This is a proposed consequence of the paper's discussion of even kernel graphs: the preceding argument rules out an even kernel graph for these values of $n$, but the conjecture's status is not established in the supplied text.