Falsifiable (the site's label; page last edited 06 December 2025): open, but a single polynomial with no short path would disprove it. The site's proof-claims tab carries one partial proof claim, the cubic case, and records no acceptance. The derived standing is claimed/disproved: the pending full claim on ani 2026 (counterexample), posted on the thread on 2026-09-07, offers a degree-seven polynomial in which, it argues, no path of length less than $2$ inside the lemniscate joins two roots; two readers reported checks, Cook formalized one member of the family in Lean, and the formal-conjectures catalog marked the problem solved with answer false on 2026-09-23, while the site labels the problem FALSIFIABLE, the community database records the problem falsifiable (status dated 2025-09-15, as of 2026-10-06), and no human review is documented. Partial claims argue the affirmative for degree three (Borisov 2026, the claim on the tab), degree four (Pendyala 2026) and trinomials (Cook 2026); two general proofs posted earlier on the thread, ani 2026 and kasko37 2026, were rejected after readers found gaps. Nothing is refereed or accepted by the site, and nothing has been built or audited in this wiki.