The fixation conjecture for noncritical majority dynamics

Let $(\eta_t)_{t\geq0}$ be median dynamics and, for an initial density $p\in[0,1]$, let $(\xi_t^p)_{t\geq0}$ be the coupled majority-dynamics process. Say that majority dynamics fixates if every vertex is eventually constant. Noncritical fixation conjecture. If median dynamics converges almost surely, then the corresponding majority dynamics fixates for every initial density $p\neq\frac12$. The preceding proposition establishes only fixation for almost every $p$ from convergence of median dynamics; this conjecture asks that the possible exceptional set be reduced to the critical density $1/2$.