Sapir's asymptotic-cone conjecture for critical exponents

Let $G$ and $G'$ be groups with Cayley graphs whose asymptotic cones are isometric. A group is non-elementary hyperbolic if it is hyperbolic and is not virtually cyclic. Sapir's conjecture. If the asymptotic cones of the Cayley graphs of $G$ and $G'$ are isometric, then their critical exponents are the same. In particular, every Cayley graph of a non-elementary hyperbolic group has mean-field valued critical exponents. The conjecture proposes that critical exponents are determined by the large-scale geometry encoded by asymptotic cones; the paper gives the consequence for non-elementary hyperbolic groups but does not establish the general claim.