Does the assumption of both Goldbach and Elliott-Halberstam conjectures imply the twin prime conjecture?

Under Goldbach conjecture, denote by $$r_{0}(n) : =\inf\{r>0,(n-r,n+r)\in\mathbb{P}^{2}\}$$ for any large enough composite integer $$n$$ . Say a positive composite integer $$n$$ is hexahedral if $$r_{0}(n)\mid 6$$. From Polymath8b, Elliott-Halberstam conjecture implies there are infinitely many pairs of primes $$(p,q)$$ with $$q-p=12$$. Denoting for such pairs by $$n_{p,q} : =(p+q)/2$$, it follows that if the latter is composite, then $$r_{0}(n_{p,q})=6$$. If it is prime, then $$r_{0}(p+3)=3$$ . In any case, we get infinitely many hexahedral integers.

Now my question is : is the density of integers $$n$$ such that $$r_{0}(n)=1$$ among hexaedral integers positive ? I think it should be equal to $$1/6$$, which may be a consequence of Chebotarev's theorem.

Numerical experiments give a proportion of around $$0.175...$$ among hexaedral integers below $$10^7$$ .