Mézard–Montanari's Kesten–Stigum conjecture for the Potts model

Consider the Potts model with $q$ symbols on a $d$-ary tree. Let $\theta=\tanh(\beta)$ and let

$$\lambda_2=\frac{e^{2\beta}-1}{e^{2\beta}+q-1}=\frac{2\theta}{q-(q-2)\theta}.$$

Here reconstruction means that information about the root persists at arbitrarily large depth, and $d_{\max}$ is the upper degree range considered in the conjecture.

Mézard–Montanari's conjecture. If $q\leq 4$ and $d<d_{\max}$, then there is reconstruction if and only if

$$d\lambda_2^2>1.$$

The conjecture asserts sharpness of the Kesten–Stigum bound in the stated range. It was based on numerical simulations; the authors reportedly expected $d_{\max}=+\infty$. Later work cited in the paper proves sharpness for $q\leq 3$ at sufficiently large degree and failure of sharpness for $q\geq 5$, leaving the small-degree cases relevant here open.