Primes obtained concatenating two primes with the same digital root respectively digital sum

In this paper I make the following three conjectures: (I) for any k having one of the values 1, 2, 4, 5, 7 or 8, there exist an infinity of primes obtained concatenating two primes that both have the digital root equal to k; (II) for any n positive integer, not divisible by 3, n ≥ 4, there exist primes obtained concatenating two primes that both have the digital sum equal to n; (III) there exist an infinity of values of n, positive integer, for which exist an infinity of primes obtained concatenating two primes that both have the digital sum equal to n.