Odlyzko–Rubinstein–Wolf conjecture on the divisibility of jumping champions

Let $p_n$ denote the $n$-th prime. For $x>0$ and an integer $d$, define

$$N(x,d)=\sum_{\substack{p_{n+1}\leq x\p_{n+1}-p_n=d}}1,$$

and let $N^*(x)=\max_d N(x,d)$. The integers $d$ for which $N(x,d)=N^*(x)$ are called jumping champions.

Odlyzko–Rubinstein–Wolf conjecture. The jumping champions tend to infinity. Furthermore, any fixed prime $p$ divides all sufficiently large jumping champions.

This conjecture predicts the eventual arithmetic structure of the most common consecutive-prime gaps; the source presents it as one of two conjectures formulated from heuristic arguments and numerical evidence.