Strong Terminating Conjecture for M&m sequences

Starting from the normalized initial values $0<x<1$, let $(x_4,x_5,\ldots)$ be the associated M&m sequence, and let $L(x)$ denote the minimum integer $k$ such that $x_n=x_k$ for all $n>k$. Strong Terminating Conjecture. For every $0<x<1$, there exists an integer $k$ such that $x_n=x_k$ for all $n>k$; equivalently, $L(x)<\infty$, or the associated sequence of medians eventually becomes constant. This is the normalized formulation of the stability conjecture. The paper's abstract says that the conjecture is proved for a subset of possible starting conditions, while the full statement remains open in the supplied source.