The odd-argument 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 $\operatorname{lg}(n)=\left[\log_2(n)\right]$ and let $\nu=\nu_2$. The integer $e$ is such that $0\le k<2^{e-1}$.

The odd-argument 2-adic valuation conjecture. If $0\le k<2^{e-1}$, then

$$\nu\bigl(f(2^e+2k+1)-f(2k+1)\bigr)\ge e-2\operatorname{lg}(k+3)+2\nu(k+1).$$

The authors present this as a potentially more tractable consequence of their study of $2$-definability. The conjecture concerns valuation growth along the odd arguments and is intended to help establish $2$-definability for additional $2$-adic integers.