Unbounded transit-time conjecture for the mean-median map

For an initial set of size three, use affine invariance to reduce to the initial set $[0,x,1]$ with $x\in\left[\frac{1}{2},\frac{2}{3}\right]$. Let $\tau(x)\in\mathbb{N}_{>3}\cup\{\infty\}$ be the transit time, meaning the time step at which the orbit stabilizes. Unbounded transit-time conjecture. The function $\tau$ is unbounded. This conjecture concerns the increasingly long transients of the mean-median map and is stated as open in the source, even though the paper proves unbounded transit time for a slightly modified recursion.