Open, in the site's label (OPEN; page last edited 23 March 2026), which attaches to the pair of questions, listed in the frontmatter as the parts weak_bound (the bound $n^{2+o(1)}$) and strong_bound (the bound $\ll_k n^2$). The second question is answered no: for $k\le2$ the bound is trivial, and for every $k\ge3$ the product exceeds $c\,n^2\log n$ infinitely often, by a Pell-equation construction of van Doorn completed by Tao with Gemini Deepthink in the problem's thread on 2025-11-20, which the site's commentary credits; the corpus records it as the pending partial claim van Doorn 2025 and Hughes's sharpening of the rate as Hughes 2026, both claimed, since the site labels the problem OPEN and neither has a refereed publication or Lean the corpus built. The first question is open unconditionally; Hughes's conditional yes under a consequence of the abc conjecture is his conditional claim.