The eventual structure of non-terminal residue classes for Stirling-number valuations

Let $S(n,k)$ denote the number of partitions of an $n$-element set into exactly $k$ nonempty subsets, and let $v_2$ denote the $2$-adic valuation. For fixed $k\in\mathbb{N}$, consider the residue classes

$$C_{m,j}^{(k)}=\{2^m i+j:i\in\mathbb{N}\text{ and }2^m i+j\geq k\}.$$

A class is non-terminal if $v_2(S(n,k))$ is not constant on it, and terminal otherwise; the parameter $m$ is the level of the class. The eventual non-terminal-class conjecture. For every fixed $k\in\mathbb{N}$, there exist a level $m_0(k)$ and an integer $\mu(k)$ such that, for every $m\geq m_0(k)$, the number of non-terminal classes of level $m$ is $\mu(k)$, independently of $m$. Moreover, for each $m\geq m_0(k)$, every one of these $\mu(k)$ non-terminal classes splits into one terminal subclass and one non-terminal subclass, with the latter generating the set of classes at the next level.

The conjecture proposes an eventual stable branching pattern for the $2$-adic valuations of Stirling numbers when $k$ is fixed. The source gives the explicit valuations for $1\leq k\leq4$ but does not provide a resolution of the general assertion.