Prime p₀= 4387431178746572711 starts a Tuple of 14 consecutive primes, which form 7 twin primes (TPT).
d_n: 0 2 58 2 64 2 22 2 46 2 10 2 28 2 (?)
a_n: 0 2 60 62 126 128 150 152 198 200 210 212 240 242
All 14 integers p_n = p₀+a_n = p_n-1+d_n are prime and there are no other primes in the gaps.
diameter: 242
Found by: [SG]KidDoesCrunch
db: batch=85, k=14, 58 64 22 46 10 28

SPT(16)	4387429545100460059	0 28 34 138 168 220 234 264 274 304 318 370 400 504 510 538	-1633646112652
TPT(14)	4387429628091172589	0 2 78 80 132 134 192 194 372 374 378 380 498 500	-1550655400122
SPT(16)	4387430304304067291	0 36 42 60 62 96 108 162 206 260 272 306 308 326 332 368	-874442505420
STPT(10)	4387430434229573741	0 2 6 8 48 50 90 92 96 98	-744516998970
SPT(16)	4387430760098934817	0 10 66 70 90 114 142 154 282 294 322 346 366 370 426 436	-418647637894
SPT(16)	4387431093006193667	0 14 26 50 54 84 96 110 186 200 212 242 246 270 282 296	-85740379044
STPT(10)	4387431781887825659	0 2 18 20 30 32 42 44 60 62	603141252948
SPT(16)	4387432035420131389	0 4 30 42 54 102 118 124 258 264 280 328 340 352 378 382	856673558678
SPT(16)	4387432257998258059	0 24 34 90 108 132 142 148 204 210 220 244 262 318 328 352	1079251685348
STPT(10)	4387432285637323577	0 2 30 32 72 74 114 116 144 146	1106890750866
STPT(10)	4387432710975172679	0 2 72 74 120 122 168 170 240 242	1532228599968