The balanced-power narrower-board conjecture for Maker-Breaker crossing games

Let $S_{m\times n}$ denote the $m\times n$ square grid, and let the $(p,p)$-crossing game be the Maker-Breaker game in which Maker and Breaker claim equal numbers of edges per round. Balanced-power narrower-board conjecture. For every $\varepsilon>0$ and every $p\in\mathbb{N}$, there exists $m_0\in\mathbb{N}$ such that, for all $m\geq m_0$, Breaker wins the $(p,p)$-crossing game on

$$S_{m\times\lceil(1-\varepsilon)m\rceil}.$$

This conjecture concerns overcoming Maker's first-player advantage when the players' powers are balanced; the statement is presented as an open problem in the paper.