Wojciechowski–Keller volume-boundary conjecture for stochastic completeness

Let $(V,E)$ be a graph, choose a fixed root $x^*\in V$, and let $B_r$ be the ball of radius $r$ about $x^*$. Let $\partial B_r$ denote its outer vertex boundary, and write $\#A$ for the cardinality of a finite set $A$. Wojciechowski–Keller conjecture. If

$$\sum_{r=0}^{\infty}\frac{\#B_r}{\#\partial B_r}=+\infty,$$

then $(V,E)$ is stochastically complete.

This conjecture gives a volume-to-boundary criterion for stochastic completeness. The supplied status evidence says that Bär and Bessa constructed a counterexample to Grigor'yan's conjecture; accordingly, this statement is refuted.