Prime p₀= 4387487644482081239 starts a Tuple of 14 consecutive primes, which form 7 twin primes (TPT).
d_n: 0 2 58 2 178 2 16 2 10 2 10 2 94 2 (?)
a_n: 0 2 60 62 240 242 258 260 270 272 282 284 378 380
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: 380
Found by: [SG]KidDoesCrunch
db: batch=85, k=14, 58 178 16 10 10 94

SPT(16)	4387485501048976897	0 42 82 90 124 126 130 132 244 246 250 252 286 294 334 376	-2143433104342
SPT(16)	4387486354764882731	0 36 62 108 216 218 246 300 428 482 510 512 620 666 692 728	-1289717198508
SPT(16)	4387486660941566719	0 12 34 60 78 82 154 162 220 228 300 304 322 348 370 382	-983540514520
SPT(16)	4387486754114947393	0 16 66 96 106 124 148 154 180 186 210 228 238 268 318 334	-890367133846
SPT(16)	4387486911443466101	0 8 42 62 66 96 110 182 186 258 272 302 306 326 360 368	-733038615138
STPT(10)	4387487161550980841	0 2 6 8 48 50 90 92 96 98	-482931100398
SPT(16)	4387487660688692099	0 42 60 108 134 192 228 242 270 284 320 378 404 452 470 512	16206610860
SPT(16)	4387487671681225997	0 42 90 92 132 146 186 200 246 260 300 314 354 356 404 446	27199144758
SPT(16)	4387487924777617931	0 38 90 120 126 150 176 182 186 192 218 242 248 278 330 368	280295536692
SPT(16)	4387488019543869179	0 30 38 42 60 80 92 98 102 108 120 140 158 162 170 200	375061787940
TPT(14)	4387488044407048037	0 2 42 44 72 74 120 122 132 134 270 272 450 452	399924966798