Mean-median map median convergence conjecture

Let $[x_1,\ldots,x_{n_0}]$ be a finite non-empty real multiset, and let $(x_n)_{n=1}^{\infty}$ be its mean-median map orbit. Write $\mathcal{M}_n$ for the median of $[x_1,\ldots,x_n]$, and call $(\mathcal{M}_n)_{n=n_0}^{\infty}$ the associated median sequence. Mean-median map median convergence conjecture. The median sequence of every initial set converges. The median sequence is known to be monotonic and to converge under certain repeated-point conditions, but convergence for every initial set remains open, including for initial sets of size three.