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Claim. For all but $o(x)$ integers $n\le x$,

$$\tfrac{1}{20}\log_* x\le h(n)\le H(n)\le 4\log_* x,$$

where $\log_*$ is the iterated logarithm. Hence $h(n)$ and $H(n)$ both have order $\log_* n$ for almost all $n$, and the set of $n\le x$ with $H(n)/h(n)>80$ has size $o(x)$, so the problem's second question, whether $H(n)/h(n)\to\infty$ for almost all $n$, is answered in the negative.

Argument. The upper bound is a union bound over bad pairs: consecutive chain members $d<e$ with $e\equiv1\pmod d$ are coprime, so both dividing $n$ costs a factor $1/(de)$, and excluding short steps forces a divisor chain to grow like an exponential tower, giving $H(n)\ll\log_* x$. The lower bound builds a prime chain through the windows $Y_1=\exp\exp\sqrt{\log_* x}$ and $Y_{j+1}=\exp\exp(Y_j^2)$, using a reciprocal prime-mass lemma derived from the Siegel--Walfisz theorem and partial summation, and transfers the independent-prime model to the integers $n\le x$ through the Chinese remainder theorem modulo a primorial that is $o(x)$. The write-up is the Overleaf draft linked above.

Claimant and system. The forum user Treasure42 posted the argument on 25 April 2026 as a candidate proof for independent checking and states that GPT-5.5 Pro generated the proof note and most of the write-up; the post links the generating and verifying transcripts. The disclosure, dropped while the post was being merged, was restored on 12 May 2026 after the site's curator asked for it.

Acceptance. Reviewed: the site's curator, Thomas Bloom, summarized the lower-bound and upper-bound arguments from this write-up on 11 May 2026 and records on the site (page last edited 11 May 2026) the two-sided bound $\log_* n\ll h(n)\le H(n)\ll\log_* n$ for almost all $n$, attributes the proofs to GPT 5.5, and marks the problem solved. The lower bound is a quantitative form of the argument Wouter van Doorn sketched on the thread on 14 October 2025 for $h(n)\to\infty$ almost always. No refereed publication and no Lean proof of this write-up exist; the Lean developments on the thread formalize Turturean's sharper asymptotics.