Proved. The status-defining source is the site's own account of a construction that its commentary attributes to GPT-5.2 Pro prompted by Kevin Barreto (2 March 2026): the construction gives infinitely many $n$ with every prime $p\le(2+\epsilon)\log n$ dividing the product, for each fixed $\epsilon<3/\log4-2\approx0.164$, so the constant $2$ can be raised to any constant below $3/\log4\approx2.164$, and a second run of GPT-5.2 Pro the same day proves the statement for every constant in place of $2$. Tao's thread comments (2--3 March 2026) sketch an elaboration giving infinitely many $n$ with $q(n,\log n)>\frac{1-o(1)}2\frac{\log\log n}{\log\log\log n}\log n$; a write-up of that elaboration produced by GPT-5.2 Thinking, posted by Nat Sothanaphan on 3 March 2026, states the $\gg$ form, and its proof gives the constant $\frac12$, which Tao's reply credits to GPT; it is the second claim page, Sothanaphan, claimed and not adopted. The site's curator, Thomas Bloom, accepted the answer on 7 March 2026 with the label PROVED (LEAN). No paper, preprint or refereed publication exists: the argument's written forms are documents linked from the thread on a file-sharing service (the first of them is the anonymous four-page note [Anon26], see gap (1); the second was not read) and an external Lean file at a pinned commit, which proves the formal-conjectures statement with $\epsilon=0.1$; it was not built or independently audited here, and no local kernel credit is claimed. This is a source-supported solution accepted by the site, distinct from a claim of journal refereeing: no refereed publication, no arXiv version and no independent expert review of the argument was found. The results in hand before 2026 were Erdős's own example, $n$ the product of the primes between $\log n$ and $(2+o(1))\log n$, giving $q(n,[\log n])\ge(2+o(1))\log n$ once $n+1$ rather than $n$ is that product or the product starts at $i=0$ (see the origins paragraph below), and his crude bound $q(n,k)<(1+o(1))k\log n$ (both 1979). The standing is derived from the claim page the construction's claim page (Barreto, 2026) with these qualifications (accepted on the site's documented acceptance; no paper, no refereed publication and no formal audit here).