Perfect-power maximality conjecture for periodic grid quotients

Let $d\ge2$ and $n\ge3$. Let $\Lambda\subset\mathbb{Z}^d$ be a full-rank sublattice, let $S\subset\mathbb{Z}^d$ be $\Lambda$-periodic, and let $\mathcal L_d[S]/\Lambda$ be the corresponding finite simple quotient graph. Suppose this quotient is connected and has $n^d$ vertex orbits.

Periodic perfect-power maximality conjecture. Its spanning-tree count is at most

$$\tau(\mathcal L_d[S]/\Lambda)\le \tau(C_n^{\square d}).$$

Equality should occur only for the standard product torus $C_n^{\square d}$, up to the natural quotient symmetries. The conjecture extends the comparison for rectangular tori to general periodic grid quotients, but the appropriate ambient class is less canonical than the class of connected induced subgraphs of $\mathbb{Z}^d$, so this remains open.