Benjamini–Chan–O'Donnell–Tamuz–Tan conjecture on agreement in majority dynamics

Let $G(n,p)$ be the Erdős–Rényi random graph on vertex set $[n]$, with each edge present independently with probability $p$. For each vertex $v\in[n]$, let $s_0(v)\in\{\pm1\}$ be sampled uniformly and independently, and let $\varepsilon\in(0,1]$. Under majority dynamics, each opinion $s_t(v)$ updates synchronously to the majority opinion among the neighbors, retaining its previous value in case of a tie. An $(1-\varepsilon)$-proportion agreement means

$$\left|\sum_v s_t(v)\right|\geq (1-2\varepsilon)n.$$

Benjamini–Chan–O'Donnell–Tamuz–Tan conjecture. With probability $1-\varepsilon$, the vertices in $G(n,p)$ have an $(1-\varepsilon)$-proportion agreement after sufficiently many days whenever $p=\omega(1/n)$.

The conjecture concerns whether sparse Erdős–Rényi graphs typically drive a uniformly random initial opinion configuration close to consensus. The supplied text gives no evidence of resolution, so its status remains open.