Benjamini–Chan–O'Donnell–Tamuz conjecture on limiting majority-dynamics cycles

Let $G(n,p)$ be a random graph with vertex set $V_n$, let $d=np$, and let $S_t(v)\in\{-1,+1\}$ be the state of vertex $v$ after $t$ rounds of synchronous majority dynamics. Benjamini–Chan–O'Donnell–Tamuz conjecture. With high probability over the random graph and the initial state, for every sufficiently large $n$ the following hold: if $d\to\infty$, then for every $\varepsilon>0$,

$$\lim_{t\to\infty}\left|\sum_{v\in V_n}S_{2t}(v)\right|\in[(1-\varepsilon)n,n];$$

if $d$ is bounded, then for every $\varepsilon>0$,

$$\lim_{t\to\infty}\left|\sum_{v\in V_n}S_{2t}(v)\right|\in[(1-\varepsilon)n/2,(1+\varepsilon)n/2].$$

The first regime predicts eventual two-cycles near unanimity, while the bounded-degree regime predicts two-cycles with approximately equal populations of the two states. The paper verifies the first regime under a sufficiently rapid growth condition on $d$, but leaves the full conjecture open.