The 2-adic valuation conjecture for Stirling-number-related quantities at powers of two

Let $e_2(k,n)$, $\overline e_2(n)$, and $s_2(n)$ be the quantities defined in the paper, with

$$\overline e_2(n):=\max\{e_2(k,n):k\in\mathbb Z\}.$$

Here $\nu$ denotes the 2-adic valuation, with the convention $\nu(0)=\infty$.

Power-of-two valuation conjecture. If $n=2^t$, then

$$\overline e_2(n)=e_2(n-1,n)=s_2(n)+1,$$

while if $n=2^t+1$, then

$$\overline e_2(n)=e_2(n-1,n)+1=s_2(n)+1.$$

These identities predict the exact maximum of the relevant 2-adic valuations for the two families $n=2^t$ and $n=2^t+1$, and are motivated by the paper's analysis of divisibility properties of Stirling numbers. The paper describes an approach toward proving them; no resolution is supplied here.