Selberg's 1943 result on primes in short intervals and primality radius

Question

This preprint: https://arxiv.org/abs/2207.05038 states in the last paragraph of the first page that a result of Selberg (1943) implies that under RH, almost all intervals of the form $(x,x+\left(\log x\right)^{2+\varepsilon})$ contain primes. Denoting under Goldbach's conjecture by $r_{0}(n):=\inf\{r\mid r\geq 0,(n-r,n+r)\in\mathbb{P}^{2}\}$, does Selberg's result imply that under both RH and Goldbach's conjecture, almost all integers $n$ fulfill $r_{0}(n)\ll_{\varepsilon}\left(\log n\right)^{2+\varepsilon}$?