Under the Goldbach conjecture, define a primality radius of a large enough composite positive integer $ n $ to be a positive integer $ r $ such that both $ n-r $ and $n+r $ are primes. Assume $k $ is such that $1 $ is a primality radius of $ k $, that is, such that there exists a couple of twin primes whose half sum is $ k $. Does there, under these assumptions, always exist some integer $ m>k $ such that $1 $ , $ k-1 $ and $ k+1 $ are primality radii of $ m $? If yes, this would imply, under Goldbach conjecture, that there are infinitely many twin primes.
Example :$ k=42 $, half sum of twin primes $ 41 $ and $ 43 $. One can take $ m=60 $ to get the twin primes $ 17 $ and $ 19 $ , $ 59 $ and $ 61 $, $ 101 $ and $ 103$.
In case of an affirmative answer to my question, I'd also be interested in an upper bound of the smallest such $ m $ in terms of $ k $.