Conjecture on error probability near the broadcasting threshold

Let $P_e(\delta)$ denote the optimal reconstruction error probability in the binary-tree broadcasting model, let $\delta_c$ be the critical channel parameter, and write $\delta=\delta_c-\tau$ with $\tau\ll 1$. Error-probability critical exponent conjecture.

$$P_e(\delta_c-\tau)=\frac12-\Theta(\sqrt{\tau}).$$

This conjecture identifies the unresolved critical exponent for reconstruction error near the threshold; the available bounds only place its behavior between square-root and linear corrections.