An unusual conjecture on primes involving concatenation and repunits

In this paper I make the following conjecture: for any k positive integer there exist an infinity of primes p such that the number q, obtained concatenating (p – k) with p then, repeatedly k times, with the digit 1, is prime. Examples: for k = 1, there exist p = 19 such that q = 18191 is prime; for k = 2, there exist p = 5 such that q = 3511 is prime; for k = 3, there exist p = 7 such that q = 47111 is prime; for k = 4, there exist p = 37 such that q = 33371111 is prime; for k = 5, there exist p = 11 such that q = 61111111 is prime; for k = 6, there exist p = 17 such that q = 1117111111 is prime.