Open. The site labels the problem OPEN (page last edited 17 April 2026, as of 2026-10-06). Its commentary records that $F(x)\to\infty$, with the elementary bound $F(x)\gg\log x$ from van Doorn's comment, the lower bound $F(x)\ge\exp((\log x)^{3-o(1)})$ from the construction of Balister, Bollobás, Morris, Sahasrabudhe and Tiba, and the trivial upper bound $F(x)\le\exp(O(x\log x))$; the lower bound is an accepted partial claim on its claim page (Balister, Bollobás, Morris, Sahasrabudhe and Tiba, 2019). The proof-claims tab carries one full claim, recorded without adoption on Snyder's claim page (2026): $\log\log F(x)/\log x\to1$, that is $F(x)=\exp(x^{1+o(1)})$, placing $F$ near the trivial upper bound, with a Lean 4 proof in a downloadable bundle produced by the Star Fleet Math system running GPT 5.6 in a custom harness (accepted in Star Fleet Math's listing on 2026-07-12 and submitted to the site on 2026-07-15 by Colin Snyder). The standing is claimed through that pending full claim; this corpus has not built the Lean bundle, and no outside review is recorded.