The site's label is FALSIFIABLE, which the site explains as an open problem that a finite counterexample could settle; it is recorded here as the site's label, not as the standing. The standing derives from the claim pages. The full claim of 5 September 2026 on the site's proof-claim tab, Gessel's counterexample, asserts a disproof: the statement is a universal statement over $A$, $n$ and $m$, and the claim exhibits a counterexample built on the $257$-smooth integers in $(T,256T]$ at a scale $T=256^k$ with $k<4096$ that its pigeonhole argument guarantees but does not name. The site has not reviewed or acted on that claim (its label and the commentary of 8 April 2026 were unchanged on 2026-09-18), its Lean file is neither built nor audited here, and no referee or named expert has examined it, so the claim is claimed and the problem's standing is claimed, disproved. The four partial claims, of 20 March 2026, Chojecki's note for sets with at most three primitive elements, excess at most five or at most nine covered integers up to $n$; of 30 April 2026, MalekZ's note for the three-element family $\{2u,3v,uv\}$; of 27 August 2026 (submitted to the tab on 28 August), Ewing's candidate proof for sets with at most seven primitive elements; and of 5 September 2026 (submitted to the tab on 29 September), the shoal-rat project's proof for primitive sets whose covered integers up to $n$ exceed the members by at most $15$, are positive results for restricted classes consistent with the counterexample and derive nothing. No refereed source proving or disproving the statement was found in the search whose scope the Current assessment records; the positive results in hand are forum items for restricted classes of $A$ (two-element sets, primitive sets containing $2$, sets of primes in the limit $m\to\infty$ with an unspecified constant), consistent with the counterexample, whose set is neither of those.