Multipartite universal-cover ball spectral-radius convergence conjecture

Let $D$ be a multipartite degree matrix, let $T_D$ be its universal cover, and let $T_{D,r}(s)$ denote the radius-$r$ ball around $s\in V(T_D)$; write $\rho(\mathord\cdot)$ for spectral radius. Convergence-rate conjecture. There exists a constant $c=c(D)$ such that for every $s\in V(T_D)$,

$$\rho(T_{D,r}(s)) \ge \rho(T_D)-cr^{-2}.$$

This conjectures an $O(r^{-2})$ rate for convergence of the spectral radii of finite balls in multipartite universal covers to the spectral radius of the cover. It generalizes the rate established in the preceding theorem for regular trees; no resolution is supplied here.