Maximum volume growth conjecture for products of regular trees

Let $T_p$ be the infinite $p$-regular tree, and let $T_p^q$ be its $q$-fold Cartesian product, which is $pq$-regular. For a graph $G$ and vertex $x$, write $\mathrm{dist}^{-1}_x(i)$ for the vertices at graph distance $i$ from $x$, and let $\kappa$ denote the graph curvature. Maximum volume growth conjecture. Let $G$ be a $pq$-regular graph such that, for all $u,v\in V(G)$, $\kappa(u,v)\geq \frac{2-p}{pq}$. If $x\in V(G)$ and $y\in V(T_p^q)$, then for every $i\geq 0$,

$$|\mathrm{dist}^{-1}_x(i)|\leq |\mathrm{dist}^{-1}_y(i)|.$$

The conjecture asserts that $T_p^q$ has maximal volume growth among $pq$-regular graphs with the specified curvature lower bound. In particular, the $d$-dimensional lattice $T_2^d$ is conjectured to have the fastest volume growth among $2d$-regular graphs with curvature lower bound $0$; this is presented as a graph analogue of the Bishop volume comparison theorem.