SKIP: The live page is marked OPEN but lists 1 partial claimed proof (Liam Price using GPT-5.6 Sol Pro, submitted 2026-07-18) asserting that every infinite sum-free sequence satisfies \(\limsup_{n\to\infty}(a_{n+1}-a_n)/n=\infty\), so the requested no-duplication stop condition applies; “Currently working on this problem” is None.