The refined distribution conjecture for Pratt-tree height

For a prime $p$, let $H(p)$ be the height of its Pratt tree, let $\log_2 p=\log\log p$ and $\log_3 p=\log\log\log p$, and write $\pi(x)$ for the number of primes at most $x$. Refined Pratt-tree height conjecture. There is a quantity $E(p)$ and fixed constants $c,c'>0$ such that

$$H(p)={\rm e}\log_2 p-\frac32\log_3 p+E(p),$$

with, for every $z\geqslant 0$, the number of primes $p\leqslant x$ satisfying $E(p)\geqslant z$ bounded above and below by constant multiples of the indicated exponential tails, and with the lower tail satisfying

$$\#\{p\leqslant x:E(p)\leqslant-z\}=O\left(\exp\{-{\rm e}^{cz}\}\pi(x)\right).$$

More precisely, the upper-tail bounds are $\gg {\rm e}^{-c'z}\pi(x)$ and $\ll {\rm e}^{-cz}\pi(x)$. The conjecture predicts a tight, asymmetric distribution for the centered Pratt-tree height; the paper states that the corresponding features are proved for its probabilistic model, but not for the arithmetic problem.