The unimodality conjecture for median-dynamics marginals

Let $p_t^x$ denote the density of the marginal law of the median-dynamics opinion $\eta_t(x)$ on $[0,1]$. Unimodality conjecture. The density $p_t^x$ is symmetric around $\frac12$ and monotone non-decreasing on $[0,\frac12]$. Thus the marginal distribution should become increasingly concentrated around the central opinion; the paper presents this as an open conjecture.