The extra-power epsilon-longer-board conjecture for Maker-Breaker crossing games

Let $S_{m\times n}$ denote the $m\times n$ square grid, and let the $(p,q)$-crossing game be the Maker-Breaker game on this grid in which Maker claims $p$ edges for every $q$ edges claimed by Breaker. Extra-power epsilon-longer-board conjecture. The following hold:

1. For every $q\in\mathbb{N}$, there exists $\varepsilon>0$ such that, for all sufficiently large $n$, Maker wins the $(q+1,q)$ game on

$$S_{\lceil(1+\varepsilon)n\rceil\times n}.$$

2. For every $p\in\mathbb{N}$, there exists $\varepsilon>0$ such that, for all sufficiently large $m$, Breaker wins the $(p,p+1)$ game on

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

This conjecture proposes that giving one player extra power permits a winning strategy on a board longer than the balanced board; determining these thresholds is open beyond the special cases treated in the paper.