The larger grid graph conjecture for Trail Trap

Let $P_m$ and $P_n$ be paths on $m$ and $n$ vertices, respectively, and let $P_m\square P_n$ be their Cartesian product, the $(m,n)$ grid graph. Let ${\rm P}{2}$ denote the second player in Trail Trap. A graph is ${\rm P}{2}$-win if the second player has a winning strategy. Larger grid graph conjecture. For any $m,n\geq 3$, the grid graph $P_m\square P_n$ is ${\rm P}{2}$-win. The paper proves several parity-based cases, but the full assertion for all $m,n\geq 3$ remains open.