The complex-fugacity Glauber spectral-radius conjecture

Let $G$ be a finite graph of maximum degree $\Delta$, let $P$ be the Glauber transition matrix at complex fugacity $w$, and let $\operatorname{spr}(P)$ denote its spectral radius on the quotient by constant functions. Complex-fugacity mixing conjecture. There exists a complex domain $D_\Delta$ containing

$$[0,1/(\Delta-1))$$

such that $\operatorname{spr}(P)<1$ for every graph $G$ of maximum degree $\Delta$ whenever $w\in D_\Delta$. There are separate equal- and unequal-fugacity versions; the conjecture would provide a uniform complex-domain criterion for nonvanishing of $Z(w)$, but it remains open in the source.