Folklore equality conjecture for the maximal prime gap constant

Let $p_n$ be the $n$th prime, let $\gamma$ be the Euler–Mascheroni constant, and let $f^{-}(2)$ be the function defined in the source. The source establishes the relation

$$\limsup_{n\rightarrow\infty}\frac{p_{n+1}-p_n}{\log^2 p_n}=\frac{1}{f^{-}(2)}\geqslant 2e^{-\gamma}.$$

Folklore equality conjecture. The inequality is in fact an equality:

$$\frac{1}{f^{-}(2)}=2e^{-\gamma}.$$

This would identify the maximal-gap constant, whereas the source notes that Siegel zeros would instead imply $f^{-}(2)=0$ and hence a divergent limsup; the equality remains unresolved in the source.