The 2-adic binomial reciprocal-sum valuation conjecture

Let $f:{\mathbb N}\to{\mathbb Q}_2$ be defined by

$$f(n)=\sum_{k=0}^n\binom nk^{-1}.$$

Let $\alpha(n)$ denote the number of $1$'s in the binary expansion of $n$, and let $\nu=\nu_2$.

The 2-adic binomial reciprocal-sum valuation conjecture. If $0\le k<2^e$, then

$$\nu\bigl(f(2^e+k)-f(k)\bigr)\ge e-2\alpha(k)-2.$$

This is the paper's strongest conjecture; it was verified for $e\le 15$, with equality in that range exactly when $k=2^e-4$ or $k=2^e-2$. It would imply that $f(x)$ is $2$-definable whenever the number of zeros minus the number of ones in the binary expansion of the reductions $x_j$ tends to infinity, and hence in particular when the fraction of zeros in the binary expansion exceeds $1/2$.