Square-lattice toroidal phase-diagram conjecture

For a zero-temperature square-lattice Potts antiferromagnet on toroidal strips, let $q$ be the Potts parameter, let $q_0(L)$ be the relevant real crossing, and let $\ell$ denote the number of bridges. Square-lattice phase-diagram conjecture. At real $q$, the phases are: (a) $q\in(-\infty,0)\cup(q_0(L),\infty)$, characterized by $\ell=0$ and amplitude $b^{(0)}=1$; (b) $q\in(0,2)$, characterized by $\ell=1$ and amplitude $b^{(1)}=q-1$; and (c) $q\in(2,q_0(L))$, characterized by $\ell=2$ and amplitude $b^{(2)}_1=q(q-3)/2$. The claim summarizes empirical limiting-curve data for widths $2\leq L\leq12$, where $2\leq q_0(L)\leq3$; no proof or resolution is given.