An observation about Goldbach's conjecture and sexy primes

This is an observation I did about the Goldbach partitions (see Goldbach's conjecture) in Math Stack Exchange. The main point is that the Goldbach partitions of the even number n, in other words G(n), and the even number n+6, G(n+6), seem at least to share a pair of sexy primes p and p+6, or p and p-6, so p belongs to G(n) and p-6 or p+6 belongs to G(n+6). This is only the result of my computer tests, not a theoretical study. I have run a test for the first 9000 even numbers and I did not find a counter example for n greater than 8. I am very curious because for other distances like d=2,4,8 was very easy to find a quick (small n) counterexample of a pair of G(n) and G(n+d) not having a couple of d-primes (e.g. couple of twin primes when d=2, couple of prime primes when d=4, etc.) but not in the case of d=6 probably because the counterexample is greater than the 9000th even number or a test error (I am assuming I did correctly my test, but it could happen). If somebody could find a counterexample please let me know.

If there is not counterexample, I just guess that it could mean that assuming the Goldbach's conjecture is true, every Goldbach partition G(n) contains at least one prime p which is the sexy prime counterpart of a prime p+6 or p-6 belonging to G(n+6), so p is prime, n-p is prime, (p, n-p) belongs to G(n), because p+n-p = n, and (p+6, n+6-p-6) are primes and belong to G(n+6), so p+6+n+6-p-6=n+6, OR (p-6, n+6-p+6) both are primes and belong to G(n+6) because p-6+n+6-p+6=n+6. I wonder if the opposite way (if no counterexamples are found) could be also true.