Amdeberhan–Manna–Moll conjecture on 2-adic valuations of Stirling numbers

Let $k\in\mathbb{N}_+$. For a prime $p$, let $v_p$ denote the $p$-adic valuation, and for a sequence $c(n)$ of rational numbers define a congruence class $[a]_d$ to be constant when $v_p(c(n))$ is independent of $n\in[a]_d$, and non-constant otherwise. In particular, apply this to the Stirling numbers of the second kind $S(n,k)$.

Amdeberhan–Manna–Moll conjecture. Fix a number $k\in\mathbb{N}_+$. Then there exist $m_0(k)\in\mathbb{N}_+$ and $\mu(k)\in\mathbb{N}$ such that for any integer $m\geq m_0(k)$ there are exactly $\mu(k)$ non-constant congruence classes $[a]_{2^m}$ modulo $2^m$, and each non-constant class modulo $2^m$ splits into one constant and one non-constant class modulo $2^{m+1}$.

The conjecture concerns the eventual structure of the $2$-adic valuations of Stirling numbers of the second kind. It was proved for $k\leq 5$ by Amdeberhan, Manna and Moll and for $5\leq k\leq 20$ by Bennett and Mosteig; the supplied status evidence records a proof by Davis in a slightly different form for partial Stirling numbers when $k\leq 36$.