The uniqueness conjecture for the exceptional Potts-model solution

Let $q=3$, let $\theta\le\tilde{\theta}_1$, and let $\alpha=\alpha_1(\theta)$. Consider the system of equations in the variables $(u,v)$ given by. Uniqueness conjecture. The system has the sole solution

$$(u,v)=\left(\theta-\frac{\theta+2}{k},1\right).$$

This conjecture would strengthen the “if” direction in the $q=3$ case of the non-uniqueness theorem to an “if and only if” statement for every $k\ge2$; the source notes that it was proved for $2\le k\le4$.