About twin primes and their happy mothers.

Question

Let's say that a positive integer $n$ is a happy mother if $6$ divides $n$ and $(n-1,n+1)$ is a pair of twin primes. Is the difference between two consecutive happy mothers necessarily a happy mother itself?
Thanks in advance.

Answer 1

No; the numbers $312$ and $348$ are consecutive happy mothers, but their difference $36$ is not.

Answer 2

No, $30$ and $6$ are happy mothers, but $24$ is not.