Conjecture on mutual information near the broadcasting threshold

Let $I(\delta)$ denote the mutual information in the binary-tree broadcasting model, let $\delta_c$ be the critical channel parameter, and write $\delta=\delta_c-\tau$ with $\tau\ll 1$. Mutual-information critical exponent conjecture.

$$I(\delta_c-\tau)=\frac{4\sqrt{2}}{\ln 2}\tau+o(\tau)\text{ bit}.$$

This conjecture specifies the sharp first-order behavior of the mutual information as the critical point is approached from below, improving the currently stated linear upper and lower bounds.