Open, in the site's label (OPEN). Writing $F(n)$ for the prime in question, the site's commentary credits four refereed bounds, which the corpus accepts on their publication as partial claims, none determining the order of $F(n)$: Pólya [Po18] proved $F(n)\to\infty$ (claim page (Pólya, 1918)), Mahler [Ma35] proved $F(n)\gg\log\log n$ (claim page (Mahler, 1935)), Schinzel [Sc67b] observed that $F(n)\le n^{O(1/\log\log\log n)}$ for infinitely many $n$ (claim page (Schinzel, 1967)), and Pasten [Pa24b] proved $F(n)\gg(\log\log n)^2/\log\log\log n$ (claim page (Pasten, 2023)). The problem stays open: the site expects $F(n)\gg(\log n)^2$ for all $n$, and Erdős [Er76d] conjectured that for every $\varepsilon>0$ there are infinitely many $n$ with $F(n)<(\log n)^{2+\varepsilon}$.