The exponential connectivity decay conjecture for nonamenable graphs

Let $G$ be a connected, locally finite, quasi-transitive, nonamenable graph. Let $\tau_p(u,v)=\mathbb P_p(u\leftrightarrow v)$ be the two-point function and define the exponential connectivity decay threshold by

$$p_{\mathrm{exp}}(G)=\sup\{p\in[0,1]:\xi_p>0\},$$

where

$$\xi_p=-\limsup_{n\to\infty}\frac{1}{n}\log\sup\{\tau_p(u,v):d(u,v)\geq n\}.$$

Exponential connectivity decay conjecture.

$$p_c(G)<p_{\mathrm{exp}}(G).$$

It is known that $p_c(G)\leq p_{\mathrm{exp}}(G)$ for every quasi-transitive graph, while strict inequality is known in some settings and equality in others. The strict inequality is widely believed for nonamenable quasi-transitive graphs but remains open.