Robust reconstruction conjecture below the Kesten–Stigum threshold

Let $M$ be a channel and $T_b$ the $b$-ary tree. For $n$ and $m$, let $\sigma_{n,m}$ be obtained from the level-$n$ configuration $\sigma_n$ by applying $M^m$ independently to each symbol, and let ${\bf P}^i_{n,m}$ denote the conditional distribution of $\sigma_{n,m}$ given that the root has state $i$. Robust reconstruction conjecture. For all $M$ and $b$ such that

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

there exists $m$ such that

$$\sup_{i,j}\lim_{n\to\infty}D_V({\bf P}^i_{n,m},{\bf P}^j_{n,m})=0.$$

The conjecture formalizes the expectation that sufficiently strong fixed noise at the boundary destroys reconstruction below the Kesten–Stigum threshold, even in models where ordinary reconstruction may behave differently. The source does not state a resolution.