The lattice-quotient conjecture for extremal surface graphs

For $3\leq k\leq 6$, let $\mathcal{X}_{n,k,g}$ be the class of graphs considered in the source and define

$$\sigma_s(k)=\lim_{n\to\infty}\left(\sup_{G\in\mathcal{X}_{n,k,g}}\tau(G)^{1/n}\right),$$

where $g$ is fixed sufficiently large and $\tau(G)$ is the number of spanning trees of $G$. Lattice-quotient conjecture. The limit $\sigma_s(3)$ is achieved by quotients of the hexagonal lattice in $\mathbb{R}^2$, and the limit $\sigma_s(4)$ is achieved by quotients of the square lattice in $\mathbb{R}^2$.

The conjecture proposes specific periodic lattice models for the graphs asymptotically maximizing spanning-tree growth in the surface setting. The source does not state whether it has been resolved.