The critical bias ratio conjecture for the square lattice

Let $\rho(\mathbb{Z}^2)$ denote the critical bias ratio of the Maker–Breaker percolation game on the square lattice.

Critical bias ratio conjecture.

$$\rho(\mathbb{Z}^2)=1.$$

This is motivated by the heuristic that Maker claims a proportion $1/(1+\rho)$ of the edges and therefore behaves like bond percolation with parameter $p=1/(1+\rho)$; since $p_{\operatorname{c}}(\mathbb{Z}^2)=1/2$, the threshold should occur at $\rho=1$.