Potts-model reconstruction below the Kesten–Stigum bound

Consider the symmetric Potts channel $M$ on $q\geq 3$ symbols, with $b$-ary tree branching factor $b$ and second eigenvalue $\lambda_2(M)$. Potts-model below-bound conjecture. There exist $b$ and $\delta$ such that reconstruction is solvable on the $b$-ary tree, while

$$b\lambda_2^2(M)<1.$$

If true, this would show that the Kesten–Stigum bound is not the reconstruction threshold for these Potts models, and that census-only algorithms are inferior to optimal reconstruction.