Benjamini–Georgakopoulos conjecture on local indistinguishability of Cayley graphs

Let $\Gamma$ be a finitely presented group, and let $F$ be a connected, locally finite Cayley graph of $\Gamma$. A graph is $r$-locally $F$ if every radius-$r$ ball in it is isomorphic to a radius-$r$ ball in $F$.

Benjamini–Georgakopoulos conjecture. There exists $r\in\mathbb{N}$ such that, whenever a graph $G$ is $r$-locally $F$, $F$ covers $G$.

This conjecture proposes a general structure theorem for graphs locally indistinguishable from a Cayley graph. It was disproved by de la Salle and Tessera; the additional assumption discussed immediately before the conjecture is necessary for the positive results that followed.