Has this conjecture been proved?
Taking, for example all the prime numbers from 3 to 100,000,000
The total number of all the gaps equal to either 2 or 8 or 14 or 20...etc is 1,616,471 The total number of all the gaps equal to either 4 or 10 or 16 or 22..etc is 1,616,470 and the total number of all the gaps equal to either 2 or 6 or 10 or 14....etc equals 3,206,361
as a corollary prime gaps of the form 6, 12, 18,....etc are just as numerous as those taking the form 4, 8, 12, 16 etc
Conjectures of this type, which are equivalent to looking at the distribution of congruences classes of pairs of consecutive primes, are almost certainly true but beyond our ability to prove at the moment. For a good summary of the state of the art, see this paper by Lemke Oliver and Soundararajan.