The marginal monotonicity conjecture for median dynamics

Let $G$ be a graph, let $x\in G$, and let $(\eta_t)_{t\geq0}$ denote median dynamics with i.i.d. uniform initial opinions on $[0,1]$. For $\alpha\in[0,1]$, define $\mu_t^x(\alpha)=\mathbb{P}[\eta_t(x)\in[0,\alpha))$. Marginal monotonicity conjecture. For every $\alpha\leq\frac12$, the function $t\mapsto\mu_t^x(\alpha)$ is monotone non-increasing. This formalizes the expected movement of marginal opinions toward the center under median dynamics; the conjecture is presented as open.