Monotonicity of Potts reconstruction with the number of symbols

Consider symmetric channels $M_1$ and $M_2$ of the Potts type on $q_1$ and $q_2$ symbols, respectively, with $q_1<q_2$, and suppose

$$0<\lambda_2(M_1)=\lambda_2(M_2).$$

Potts monotonicity conjecture. If reconstruction is solvable for $M_1$, then it is also solvable for $M_2$.

This conjecture proposes that, at fixed positive second eigenvalue, reconstruction becomes easier as the number of Potts symbols increases. The source gives no resolution.