This question is somehow a follow-up from Would the following conjectures imply Cramer's conjecture? Let $g_{n,k}$ denote the quantity $p_{n+k}-p_{n}$, $s_{n,k}$ denote the quantity $p_{n+k}+p_{n}$ and $H_{k}$ denote the quantity $\lim\inf_{n\to\infty}g_{n,k}$ where $p_{m}$ is the $m$-th prime number.

I have two questions:

1) Is there a unique (possibly infinite) $a_{0}$ such that: $0<\lim \sup_{n\to\infty}\frac{g_{n,k}}{H_{k}\log^{a_{0}}(\frac{s_{n,k}}{2})}<\infty$?

2) Suppose $a_{0}$ is finite. Is $a_{0}$ necessarily equal to $\sigma_{+}$?

Thanks in advance.

EDIT December 8th 2013: If I'm not mistaken, proving unicity and finiteness of $a_{0}$ would imply $r_{0}(n)=O(\log^{a_{0}}n)$, so that such a proof could shed a light on asymptotic Goldbach's conjecture.