Open. The site labels the problem OPEN (page last edited 1 April 2026), its remarks recording the bounds of [APSSV26] and of Tang and ChatGPT and crediting no solution. The standing derives from the claim pages, three pending partial claims, none of which determines the order of $f(n)$: Tang 2026, a research note with the bound $f(n)\le\lceil n^{12/17+\varepsilon}\rceil$ and the thread's exponent $30/43$; Alexeev, Putterman, Sawhney, Sellke and Valiant 2026, an arXiv preprint with $f(n)\le(24/(\pi^2-6)+o(1))(\log n)^2$ for all large $n$ and $f(n)\ge(1/2+o(1))\log n$ along a sequence; and Bae 2026, an arXiv preprint with a Lean 4 proof that $f(n)\ge(1/2-o(1))\log n\,\log\log n/\log\log\log n$ for infinitely many $n$, which the site's curator relabeled partial because the order of $f(n)$ stays undetermined. So the problem is open with no settling or pending full claim. Known Results below record the density-one results, which have no claim pages.