Tree reconstruction conjecture for the two-state broadcast process with revealed nodes

Let $T$ be an infinite tree with root $\rho$. In the two-state broadcast process, each vertex label $\tau_v\in\{\pm 1\}$ is generated by assigning the root a uniformly random label, then propagating labels along each edge while retaining the parent label with probability $1-\eta$ and flipping it with probability $\eta$. Independently, each vertex is included in a revealed set $R$ with probability $p$, and let $\partial T_r$ be the leaves at depth $r$. Tree reconstruction conjecture. For every $p>0$ and $\eta<1/2$,

$$\lim_{r\to\infty}\mathbb{E}\left|\Pr[\tau_\rho=1\mid\tau_R]-\Pr[\tau_\rho=1\mid\tau_R,\tau_{\partial T_r}]\right|=0.$$

This conjecture would imply the claimed local optimality for the labeled two-cluster model through the tree coupling. The source reports simulations supporting it, but gives no proof; the problem remains open.