The monotonicity conjecture for the Potts functions $\alpha_m$

Let $q\geq2$, let $m\in\{1,\dots,q-2\}$, and let $K_m^*(\theta)$ be the function defined from the Potts fixed-point system. For $\theta>\theta_m$, set

$$\alpha_m(\theta)=\frac{\ln K_m^*(\theta)}{\ln\theta}.$$

Monotonicity conjecture. For each $m=1,\dots,q-2$, the function $\theta\mapsto\alpha_m(\theta)$ is monotone increasing. This would give the monotonicity observed in the plotted examples; the paper proves the related monotonicity of $K_m^*(\theta)$ but leaves the normalized logarithmic functions as conjectural.