I got this statement by upcoming mathematician Prof. Gandhi from BITS:
"All twin primes from $17$ who are the smallest elements of a pair of twin primes, can be rewritten as: $(a + b + 1)$ such that $a$, $(a+2)$, $b$ and $(b+2)$ are primes".
I checked with my own examples, $17 = 5+11+1$; $29 = 11+17+1$; 41=11+29+1; $59 = 29+29+1=17+41+1$, $71=11+59+1$
So far I have not found any counterexamples.
Could you discuss or comment on cited conjecture? I am looking for discussion.