Eswarathasan–Levine conjecture on the finiteness of divisible harmonic numbers

Let $H_n=1+\frac12+\cdots+\frac1n$ be the $n$th harmonic number. For a prime $p$, let $\nu_p(a)$ denote the $p$-adic valuation of $a$, and define

$$J_p:=\{n\geq 1:\nu_p(H_n)\geq 1\}.$$

Thus, $J_p$ consists of those $n$ for which $p$ divides the numerator of $H_n$ in lowest terms.

Eswarathasan–Levine conjecture. The set $J_p$ is finite for all primes $p$.

Eswarathasan and Levine computed $J_p$ for $p=3,5,7$ and found it finite in each case. Boyd later established finiteness for all primes $p\leq 547$, apart from possibly $p\in\{83,127,397\}$; the general conjecture remains open.