Poisson Tail Conjecture for gaps between consecutive primes

Let $p_n$ denote the $n$th prime, and let $\varepsilon>0$. Poisson Tail Conjecture. For $1\leq H\leq(\log X)^{2-\varepsilon}$, the two gap-counting quantities satisfy

$$\sum_{\substack{p_{n+1}\leq X \\ p_{n+1}-p_n\geq H}}1\asymp e^{-H/\log X}\frac{X}{\log X},$$ $$\sum_{\substack{p_{n+1}\leq X \\ p_{n+1}-p_n\geq H}}(p_{n+1}-p_n)\asymp\left(1+\frac{H}{\log X}\right)e^{-H/\log X}X.$$

For $H>(\log X)^{2+\varepsilon}$ and $X$ sufficiently large, both quantities vanish. This conjecture models prime gaps as having Poisson tails at the scale of the average gap $\log X$; the stated ranges and asymptotics are not currently known in full.