The almost-sure coin-median convergence conjecture for majority dynamics with coins

Let $\operatorname{MD}_{\mathrm{coins}}$ denote the coin-valued median dynamics process on $\mathbb{Z}^d$, and let $\eta_t(\underline{0})$ be the opinion at the origin. Coin-median convergence conjecture. For every $d\geq2$,

$$\eta_t(\underline{0})\xrightarrow[t\to\infty]{\mathrm{a.s.}}\frac12.$$

The paper explains that this is a reformulation of the zero-temperature Glauber-dynamics conjecture and that it would give convergence to the central value despite the corresponding density-$1/2$ binary process failing to converge.