Cramér–Granville conjecture for good-prime gaps

Let $\mathcal{P}_{\mathrm{good}}=\{g_1,g_2,\ldots\}$ be the subset of good primes, enumerated in increasing order, where good primes are the primes outside the paper's exceptional set. Cramér–Granville conjecture for good primes. For some $\eta>1$,

$$\limsup_{j\rightarrow\infty}\frac{g_{j+1}-g_j}{(\log g_j)^2}=\eta.$$

The paper motivates this as a strengthening of the ordinary Cramér–Granville conjecture because good primes have density one among all primes. Whether density one is sufficient to preserve these fine prime-gap statistics is not known, so the conjecture remains open.