Via the Polymath effort (circa 2014, built upon Yhang effort), we know some n<246 exists where there is an infinite count of primes of gap n.

We also know the jumping champions essentially are 4 and the primorials.

This implies n comes from the small set of {2,4,6,30,210}.  If it turns out to be 2 (or 246 can be narrowed to <3), the Twin Prime conjecture is resolved.

There are a host of OEIS entries for where both p & p+c are prime, where c= 2,6,30,210....

Here is quick PariGP code I wrote that identifies the candidate jumping champions:  primeGapSearch_faster.gp

Realize my limited computational capability (laptop, 16Gb ram).  Note that the script makes no use of the primorials, rather they fall out from the search.

Here are partial results after ~ 2 days of running:

(17:54) gp > \r  primeGapSearch_faster.gp
(17:54) gp > default(primelimit,4290000001)
(17:54) gp > \p 4
   realprecision = 19 significant digits (4 digits displayed)
(17:54) gp > default(parisize,2000000000)
  ***   Warning: new stack size = 2000000000 (1907.349 Mbytes).
(17:55) gp > genit()
candidate= 6  2.213
candidate= 30  3.097
candidate= 210  3.822
candidate= 2310  4.324
candidate= 30030  4.747