Twin prime conjecture (Goldbach-Collatz remix)

Question

Assuming Goldbach's conjecture, let's denote $r_{0}(n)$ for any integer $n$ greater than $1$ the smallest non negative integer $r$ such that both $n+r$ and $n-r$ are primes. Let $f$ be the map $m\mapsto m.r_{0}(m)$ if $m>1$ and $m\mapsto 0$ otherwise. Is it true that there are infinitely many twin primes if and only if for all non negative integer $m$, the sequence $(u_{n}(m))$ with $u_{0}(m)=m$ and $u_{n+1}(m)=f(u_{n}(m))$ is bounded?
Thanks in advance.