Optimal Power of Few conjecture for majority dynamics

Let $G(n,p)$ be an Erdős–Rényi random graph, let $\Delta$ denote Red's initial advantage, and let $p$ satisfy

$$p \ge (1 + \lambda)(\log n)/n.$$

Optimal Power of Few conjecture. There is a function $f(x)$ such that $f(x)>1/2$ for $x>0$, $f(x)$ tends to $1$ as $x$ tends to infinity, and, in majority dynamics on $G(n,p)$ with initial advantage $\Delta$ for Red, Red wins with probability

$$f(\Delta\sqrt{p}).$$

This formalizes the claim that an initial advantage of order $p^{-1/2}$ is sufficient for Red to win with probability bounded away from $1/2$ and that the winning probability approaches $1$ as the scaled advantage grows. A result of Sah and Sawhney, improving work of Devlin and Berkowitz, confirms the conjecture when $p$ tends to zero sufficiently slowly, namely when $p\ge (\log n)^{-1/16}$; the general case remains open.