Solved. The site labels the problem SOLVED (page last edited 1 September 2026) and credits GPT Pro, prompted by Price, for a proof that infinitely many $n$ have $\binom{2n}{n}$ and $\binom{2n+2}{n+1}$ with the same prime divisors, at least a constant times $x^{1/2}$ of them below $x$ for all large $x$; the result is recorded on the claim page Price 2026, and the site's proof-claim entry states that the proof has been accepted as correct. The question is a yes-or-no question answered yes, so the claim value is proved. A third-party Lean formalization of the argument is registered with the Palomar registry and copied into Boris Alexeev's repository of formalized Erdős problems; this corpus has built neither, and nothing is refereed. The standing in the frontmatter derives from the claim page.