Quadruple Lucky Numbers

It's known that the frequency of lucky numbers is similar to that of prime numbers. The frequency of twin lucky numbers is also similar to that of twin primes. The similarities do not end there. Quadruple prime numbers occur in a \(p, p+2, p+6 \text{ and } p+8\) pattern. For example: 101, 103, 107 and 109 or 191, 193, 197 and 199. 

Quadruple lucky numbers follow a similar pattern: \(n, n+2, n+6 \text{ and } n+8 \). The first members of these quadruplets populate OEIS A139783:

A139783

Quadruple lucky numbers (lower terms). Numbers \(n\) such that \(n, n+2, n+6, n+8\) are all Lucky numbers.

The initial members of the sequence are (permalink):

1, 7, 67, 127, 613, 925, 1495, 1765, 2209, 2815, 3403, 5965, 6661, 8827, 9115, 15229, 16387, 18145, 19153, 21925, 23563, 24637, 27031, 27199, 28987, 31381, 32635, 34717, 35701, 36673, 40447, 43225, 43975, 47419, 50317, 51157, 56263, 64495

For example, my diurnal age today (27199) belongs to the sequence and thus 27199, 27201, 27205 and 27207 are all lucky numbers. All four numbers are composite but I got to thinking whether there are quadruple lucky numbers that are also quadruple prime numbers?

To begin with the initial primes would need to end in the digit 1 and up to one million the only candidates fail as can be seen below:

6661

6663 = 3 * 2221

6667 = 59 * 113

6669 = 3^3 * 13 * 19

27031

27033 = 3 * 9011

27037 = 19 * 1423

27039 = 3 * 9013

70351

70353 = 3^2 * 7817

70357 = 7 * 19 * 23^2

70359 = 3 * 47 * 499

125311

125313 = 3 * 41771

125317 = 113 * 1109

125319 = 3 * 37 * 1129

321721

321723 = 3^2 * 35747

321727 = 7 * 19 * 41 * 59

321729 = 3 * 107243

602221

602223 = 3 * 191 * 1051

602227 = 602227

602229 = 3 * 197 * 1019

728911

728913 = 3 * 242971

728917 = 7 * 101 * 1031

728919 = 3^4 * 8999

786661

786663 = 3^2 * 87407

786667 = 7 * 41 * 2741

786669 = 3 * 13 * 23 * 877

This is a limited range on which to make a judgement as to whether such a thing is possible and there may well be a theoretical proof that shows that it's impossible. However, that will have to do for now.