Perfect-power square-maximality conjecture for induced grid subgraphs

Let $d\ge2$ and $n\ge1$. If $S\subset\mathbb{Z}^d$ has $|S|=n^d$ and the induced graph $\mathcal L_d[S]$ is connected, then

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

Perfect-power square-maximality conjecture. Equality should occur only for the $d$-dimensional box $P_n^{\square d}$, up to lattice translation and coordinate permutation.

For $d=2$, this specializes to the square-maximality conjecture attributed to Procaccia and Tucker-Foltz. For non-perfect-power vertex counts, the expected maximizing induced grid subgraph may be a compact near-box shape rather than a product, so exact maximality remains open beyond the stated perfect-power case.