Post-first-step monotonicity conjecture for majority dynamics

Let $G_{n,p}$ be a random graph sampled independently of an arbitrary initial red-blue coloring of its vertices, and let $\varepsilon>0$. Assume

$$p\geq \frac{(1+\varepsilon)\log n}{n}.$$

After the first step of majority dynamics, compare the numbers of vertices of the two colors. Post-first-step monotonicity conjecture. With probability tending to $1$, whichever color is more numerous after the first step will have increasingly more vertices after every subsequent step, until all vertices have that color. This stronger conjecture would, together with the paper's central limit theorem, determine the value of $\varepsilon$ in the one-extra-vertex conjecture.