Broadcasting-with-revealed-nodes boundary irrelevance conjecture

Let $T$ be an infinite tree with root $\rho$. Assign the root a uniformly random label in $\{\pm1\}$ and propagate labels along edges, retaining the parent label with probability $1-\eta$ and flipping it with probability $\eta$. Let $(T,\tau)$ be the resulting labeled tree. Independently include each node in $R$ with probability $p$, and let $\partial T_r$ be the leaves at depth $r$. Boundary irrelevance conjecture. For every $p>0$ and $\eta<1/2$,

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

The source identifies this as the tree-information-flow conjecture underlying the local-recovery claim and reports simulations supporting it. It remains open in the supplied text.