Chen et al.'s unit-circle conjecture for Potts partition zeros

Let $G$ be a finite planar self-dual lattice, or let $G$ be the square lattice with free or periodic boundary conditions in the thermodynamic limit. For $q\geq 1$, introduce $v$ by

$$x=1+\frac{\sqrt{q}}{v},\qquad y=1+\sqrt{q}v.$$

The Potts partition zeros are considered in the half plane $\operatorname{Re}(v)>0$. Chen et al.'s conjecture. For finite planar self-dual lattices and for the square lattice with free or periodic boundary conditions in the thermodynamic limit, the Potts partition zeros in the $\operatorname{Re}(v)>0$ half plane are located on the unit circle $|v|=1$. The paper states that this conjecture is false in the finite case, although it concerns the proposed unit-circle location of zeros for self-dual systems.