Solved: the answer is yes. The site labels the problem SOLVED (page last edited 1 September 2026). The result is Theorem 1.1 of Chojecki's preprint of 13 July 2026 (arXiv 2609.17543, 14 July 2026), a gap-greedy construction over consecutive primes whose proof the preprint states was found by GPT-5.6 Sol; Sneiderman's note of 21 July 2026 reproves it with a sharper exponent, and Pratt's digested proof of 1 September 2026, posted as the site's proof exposition, presents the argument in full following both. The site's curator records the answer as yes and describes its provenance as an approach sketched by Tao and fleshed out by Chojecki, full proofs claimed from GPT by several people independently, and Pratt's simplified exposition. No refereed publication exists as of 2026-10-07, and no formal proof has been built in this corpus; three further AI-assisted claims stand unreviewed, and Chojecki's claimed proof of January 2026 is recorded as rejected. Claim pages: Chojecki and Sneiderman (accepted); Kielhorn, Pauwels and Sharma (claimed); Chojecki's January claim (rejected). Selfridge's construction, recorded under Problem 786, gives such a sequence of density greater than $1/e-\epsilon$ for every $\epsilon>0$.