Conjecture on the typical height of Pratt trees

For a prime $p$, let $H(p)$ be the height of its Pratt tree: the tree with root $p$, whose children are the prime factors of $p-1$, constructed recursively, stopping at $p=2$. Let $\pi(x)$ denote the number of primes at most $x$. Pratt-tree height conjecture. There exist constants $c,c'>0$ and real numbers $\{E(p):p~\mathrm{prime}\}$ such that

$$H(p)= e\log p - \frac{3}{2} \log\log p + E(p),$$

for all $z\geqslant 0$ and $x\geqslant 0$,

$$e^{-c' z} \pi(x) \ll |\{\mathrm{primes}~p \leqslant x: E(p) \geqslant z\}| \ll e^{-cz} \pi(x),$$

and

$$|\{\mathrm{primes}~p \leqslant x: E(p) \leqslant -z\}| \ll \exp(-e^{cx})\pi(x).$$

The conjecture describes the typical height of Pratt trees and is motivated by results supporting branching-random-walk heuristics; the source provides no resolution, so its status remains open.