The statistical valuation conjecture for Stirling numbers

Let $c$ be a positive integer, let $S(n,k)$ denote a Stirling number of the second kind, let $bd_2$ be the $2$-adic valuation, and let $c3_2(m)$ denote the sum of the binary digits of $m$. Statistical valuation conjecture. As $h$ tends to infinity, the proportion of integers $k$ with $0\le k\le c2^h$ for which

$$\nu_2\left(S(c2^h,k)\right)=\sigma_2(k)-\sigma_2(c)+\nu_2\left(\binom{c2^{h+1}-k}{c2^h}\right)$$

tends to $1$; explicitly,

$$\lim_{h\to\infty}\frac{1}{c2^h}\left|\left\{k:0\le k\le c2^h,\ \nu_2\left(S(c2^h,k)\right)=\sigma_2(k)-\sigma_2(c)+\nu_2\left(\binom{c2^{h+1}-k}{c2^h}\right)\right\}\right|=1.$$

The conjecture asserts that the displayed valuation formula holds for almost all $k$. The text notes that an earlier theorem covers about half of the relevant $k$ values, but gives no resolution of the statistical conjecture.