Continuity conjecture for the mean-median map limit function

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 $m(x)\in\mathbb{R}$ be the limit of the associated median sequence. Continuity conjecture for the mean-median map limit function. The function $m$ is continuous. The conjecture asks whether the limiting median varies continuously with the initial condition; the source states that it remains open, while this paper proves discontinuity for the limit function of a slightly modified recursion.