Conjecture that every word-hyperbolic group has property R

Let $G$ be a word-hyperbolic group. An automatic structure for $G$ has property R when its accepting automaton is both undirected and Perron–Frobenius, meaning that the adjacency matrix has a unique eigenvalue of maximal modulus. Property R conjecture. Every word-hyperbolic group has property R with respect to some generating set. The preceding question asks which groups admit such automatic structures; the source gives no evidence that this conjecture has been resolved.