Triangular-lattice toroidal phase-diagram conjecture

For a zero-temperature triangular-lattice Potts antiferromagnet on toroidal strips, let $q_0(L)$ and $q_2(L)$ be the relevant real crossings, and let $\ell$ denote the number of bridges. Triangular-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 $b^{(1)}=q-1$; (c) $q\in(2,q_2(L))$, characterized by $\ell=2$ and amplitude $b^{(2)}_1=q(q-3)/2$; and (d) $q\in(q_2(L),q_0(L))$, characterized by $\ell=2$ and the same amplitude $b^{(2)}_1=q(q-3)/2$. The source notes that phase (d) occurs at finite widths such as $L=5,7$ and, according to the preceding conjectures, survives in the thermodynamic limit; this remains empirical.