The universal $2^q-1$ bound for Potts-model boundary-law solutions

Let $q\ge2$, $k\ge2$, $\theta\ge1$, and $\alpha\in\mathbb{R}$. Let $\nu(\theta,\alpha)$ denote the number of positive solutions of the system of equations. Universal bound conjecture. For all $\theta\ge1$ and $\alpha\in\mathbb{R}$,

$$\nu(\theta,\alpha)\le 2^q-1.$$

The bound is known in the cases $\alpha=0$ and $k=2$, and computational experiments suggest that it holds for every $k\ge2$; the general case remains open in the supplied text.