Cramér–Granville conjecture on prime gaps

Let $\mathcal{P}=\{p_1,p_2,\ldots\}$ be the set of prime numbers, enumerated in increasing order. Cramér–Granville conjecture. For some $\kappa>1$,

$$\limsup_{j\rightarrow\infty}\frac{p_{j+1}-p_j}{(\log p_j)^2}=\kappa.$$

Cramér proposed the corresponding prime-gap heuristic, while Granville provided evidence suggesting the stronger lower bound $\kappa\geq 2e^{-\gamma}\approx 1.1229$, and computational evidence supports the conjecture. Its status remains open.