Strong terminating conjecture for the mean-median map

For an initial condition $x\in[0,1]$, let $(x_n)$ be the orbit generated by the mean-median map from the initial set $[0,x,1]$. The strong terminating conjecture. For every $x\in[0,1]$ there is an integer $\tau$ such that

$$x_{\tau+k}=x_\tau\quad\text{for all }k\in\mathbb{N}.$$

Termination means that the orbit eventually stabilises, so this conjecture asserts stabilisation for every initial condition. It is proved in specified neighbourhoods of $x=\frac12$ and remains unresolved in general.