Prime p₀= 4387632428191358117 starts a Tuple of 14 consecutive primes, which form 7 twin primes (TPT).
d_n: 0 2 88 2 22 2 88 2 94 2 22 2 16 2 (?)
a_n: 0 2 90 92 114 116 204 206 300 302 324 326 342 344
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: 344
Found by: [SG]KidDoesCrunch
db: batch=85, k=14, 88 22 88 94 22 16

SPT(16)	4387631750319078011	0 18 36 38 48 56 78 116 132 170 192 200 210 212 230 248	-677872280106
SPT(16)	4387631829812036527	0 22 40 94 96 126 144 150 166 172 190 220 222 276 294 316	-598379321590
STPT(10)	4387631941317277319	0 2 18 20 60 62 102 104 120 122	-486874080798
STPT(10)	4387631988068520797	0 2 60 62 72 74 84 86 144 146	-440122837320
STPT(10)	4387632102716526437	0 2 54 56 72 74 90 92 144 146	-325474831680
STPT(10)	4387632381484915061	0 2 36 38 78 80 120 122 156 158	-46706443056
STPT(10)	4387633305855760949	0 2 18 20 30 32 42 44 60 62	877664402832
STPT(10)	4387634190941983901	0 2 6 8 108 110 210 212 216 218	1762750625784
STPT(10)	4387634194354284689	0 2 12 14 90 92 168 170 180 182	1766162926572
SPT(16)	4387634386975532461	0 30 52 60 82 142 160 202 318 360 378 438 460 468 490 520	1958784174344
SPT(16)	4387634900195060611	0 6 18 48 60 88 96 126 130 160 168 196 208 238 250 256	2472003702494