Mean-median map stabilization conjecture

Let $[x_1,\ldots,x_{n_0}]$ be a finite non-empty real multiset, and let the mean-median map adjoin $x_{n+1}$ so that the arithmetic mean of the enlarged multiset equals the median of the preceding one:

$$x_{n+1}=(n+1)\mathcal{M}_n-\mathcal{S}_n.$$

Here $\mathcal{M}_n$ and $\mathcal{S}_n$ are the median and sum of $[x_1,\ldots,x_n]$, respectively, and $(x_n)_{n=1}^{\infty}$ is the resulting orbit. Mean-median map stabilization conjecture. The mean-median map orbit of every initial set stabilizes. This is a fundamental open question about the long-term behavior of the mean-median map; the source states that it remains open even for initial sets of size three.