Local belief propagation optimality conjecture with noisy side information

Under the binary symmetric stochastic block model with $\alpha$-noisy side information, let $p_{G_n}(\widehat{\sigma}_{\rm BP}^t)$ be the estimation accuracy of belief propagation after $t$ iterations and let $p_{G_n}^*$ be the optimal estimation accuracy. Local BP optimality conjecture.

$$\lim_{t \to \infty} \lim_{n \to \infty} p_{G_n}(\widehat{\sigma}_{\rm BP}^t )= \limsup_{n \to \infty}p_{G_n}^*$$

holds for all $a$, $b$, and $\alpha$. This asserts that iterated local belief propagation achieves asymptotically optimal estimation accuracy for every parameter choice; the paper notes that it can prove the claim in certain regimes but leaves the general statement open.