Broadcasting information conjecture for regular and Poisson Galton–Watson trees

Let $T$ be a regular tree or a Galton-Watson tree with Poisson offspring distribution, with root vertex $\rho$. Let $(T,\rho,\theta,W)$ denote the corresponding broadcast model with parameter $0<\theta<1$ and survey channel $W$, and let $P_e(W)$ be the error probability of the survey channel. The property (BI) is the broadcasting-information property considered in the paper. Broadcasting information conjecture. Property (BI) holds for $(T,\rho,\theta,W)$ for all $0<\theta<1$ and all $W$ such that $P_e(W)\ne\frac12$. This would extend the sufficient conditions established earlier, including the high-signal regimes described in the preceding corollary, to every nontrivial survey channel. The parser supplies no evidence that this conjecture has been resolved.