Open, in the site's label (OPEN; page last edited 2025-12-27), which attaches to the pair of questions. No source in the search of 2026-09-05, and nothing in the site's thread as of 2026-10-07, determines the smallest limsup. The liminf question is answered yes: Moser [Mo65] first proved $\liminf\lvert A\cap\{1,\ldots,N\}\rvert/N^{1/2}>1.06$ for every such $A$, and Cilleruelo [Ci93] and Habsieger [Ha95] independently raised the bound to $4/\pi$. The claim pages of Moser, Cilleruelo and Habsieger record these as partial claims settling the liminf part; the two journal papers are accepted on their refereed publication, and Moser's proceedings paper stays claimed. Balasubramanian and Ramana [BaRa01] improve $4/\pi$ only under a localization hypothesis on minimal complements, so their theorem decides nothing new and has no claim page. The limsup part is unsettled, so the problem stays open: van Doorn's construction and the thread's write-up of 2026-09-07 bound the infimum from above without determining it, and neither is a claim.