Census non-solvability at the Kesten–Stigum threshold

Let $M$ be a channel with second eigenvalue $\lambda_2(M)$, and let $T_b$ be the $b$-ary tree. The reconstruction problem is census-solvable when the root can be reconstructed from the empirical counts of the symbols at a level. Census non-solvability conjecture. The reconstruction problem is not census-solvable when

$$b|\lambda_2(M)|^2=1.$$

The surrounding theorem establishes the corresponding strict inequalities, while the equality case is left as a conjecture in the source.