The monotonicity conjecture for the Potts boundary functions $\alpha_\pm$

Let $k\geq2$ and $q\geq2$. Let $\theta_{\rm c}$ be the critical value, let $\alpha_\pm(\theta)$ be the two boundary functions, and define

$$\theta_0^+=1+\frac{q}{k-1}.$$

Monotonicity conjecture. The function $\alpha_-(\theta)$ is monotone decreasing for all $\theta\geq\theta_{\rm c}$, while $\alpha_+(\theta)$ is decreasing for $\theta\leq\theta_0^+$ and increasing for $\theta\geq\theta_0^+$, with unique minimum $\alpha_+(\theta_0^+)=0$. In the case $q=2$, $\alpha_+(\theta)$ should instead be monotone increasing for all $\theta\geq\theta_{\rm c}$.