Conjecture on logarithmic lower bounds for p-adic valuations of elementary symmetric harmonic sums

For integers $n\geq k\geq1$, define

$$H(n,k)=\sum_{1\leq i_1<\cdots<i_k\leq n}\frac{1}{i_1\cdots i_k},$$

and let $\nu_p$ denote the $p$-adic valuation. The logarithmic lower-bound conjecture. For any prime number $p$ and any integer $k\geq1$, there exists a constant $c=c(p,k)>0$ such that

$$\nu_p(H(n,k))<-c\log n$$

for all sufficiently large integers $n$. This conjecture proposes that the previously known lower bound for $\nu_p(H(n,k))$ is nearly optimal; the paper's results verify the expected logarithmic behavior in a special range where the base-$p$ representation of $n$ starts with that of $k-1$, while the general assertion remains open.