Open. The site's proof-claims tab carries one full proof claim, by David Turturean (naming GPT-6-Astra Pro as the system used, with earlier work by ChatGPT-5.5-Pro and Claude), submitted 2026-09-23 with an Overleaf write-up and a Lean repository: it answers the three questions no, no and yes, unconditionally, through a positive lower density for the set of $n$ with $m_n=p_n$ and a lower bound of order $X/(\log X)^{55}$ for the uniqueness primes up to $X$; an earlier manuscript of the same author, posted on 4 May 2026, answered the third question only under Dickson's conjecture. The site labels the problem OPEN (page last edited 07 October 2025), no referee or outside reviewer has accepted the argument, and the claim is recorded as pending on its claim page (Turturean, 2026); the derived standing is claimed.