Information retention conjecture for bounded-degree majority dynamics

Let ${G_n}_{n in \mathbb N}$ be a sequence of finite graphs with degrees bounded by $d$ and such that

$$\lim_{n\to\infty}|V_n|=\infty.$$

Here, $\delta(G_n,p)$ denotes the minimum probability of error when reconstructing the initial common opinion from the limiting opinions, and $p$ is the probability that an initial opinion agrees with the underlying signal, with $\frac12<p<1$.

Information retention conjecture. For every such sequence and every $\frac12<p<1$,

$$\lim_{n\to\infty}\delta(G_n,p)=0.$$

This conjecture asserts that bounded-degree majority dynamics retains enough information about the initial signal to permit asymptotically error-free reconstruction on every sequence of growing finite graphs. The source provides no resolution, so the conjecture is recorded as open.