One-extra-vertex conjecture for majority dynamics

Let $G_{n,p}$ be a random graph on $n$ vertices, and consider majority dynamics from an initial red-blue coloring with more red vertices than blue. For fixed $p\in(0,1)$, let $\varepsilon>0$ be a corresponding constant. One-extra-vertex conjecture. With probability at least $0.5+\varepsilon-o(1)$, the majority dynamics process on $G_{n,p}$ eventually makes all vertices red. This would extend the paper's result for an initial lead of three vertices to any positive initial lead, and likely requires a new approach.