The coin-median convergence-in-probability conjecture

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-in-probability conjecture. For every $d\geq2$,

$$\eta_t(\underline{0})\xrightarrow[t\to\infty]{\text{probability}}\frac12.$$

This is presented as a weaker version of the almost-sure coin-median convergence conjecture; its status is not resolved in the supplied text.