The main conjecture on 2-adic valuations of Stirling numbers

Let $S(n,k)$ denote the Stirling number of the second kind, and for fixed $k\in\mathbb{N}$ partition the integers $n\geq k$ into residue classes modulo $2^m$. A class is constant if the values of $\nu_{2}(S(n,k))$ on it consist of a single value; the $m$-level is formed by the non-constant classes modulo $2^m$. Define $m_{0}=m_{0}(k)\in\mathbb{N}$ by

$$2^{m_{0}-1}<k\leq 2^{m_{0}}.$$

Main conjecture. The first index for which there is a constant class is $m_{0}-1$: every class is part of the $j$-level for $1\leq j\leq m_{0}-2$. For every $m\geq m_{0}$, the $m$-level consists of $2^{m_{0}-2}$ classes, and each produces a single non-constant class for the $(m+1)$-level, so the number of classes at each level remains constant. The conjecture is supported by the examples in the paper, and the special case $k=5$ is established there.