Iterating $n\mapsto\phi(n)+1$ to a prime: iteration count, fibers, and densities (Erdős #409)

Consider the map $n\mapsto \phi(n)+1$, where $\phi$ is the Euler totient function, and iterate it. For $n$ that eventually reach a prime, let $F(n)$ be the number of iterations of $n\mapsto\phi(n)+1$ needed to first reach a prime. (i) How large can $F(n)$ be — find good upper bounds. (ii) Can infinitely many $n$ reach the same prime? (iii) For a fixed prime $p$, what is the density of the set of $n$ whose iteration first reaches $p$?

Acceptance. FULLY RESOLVES: rigorous answers to the core questions — a proven good upper bound on $F(n)$ (e.g. $F(n)\ll(\log n)^{O(1)}$, or the true order, with proof), together with a proof settling whether infinitely many $n$ reach the same prime and a determination of the density of the fiber of a fixed prime $p$; full written proofs. ADVANCES (each itself checkable): a proven nontrivial upper bound on $F(n)$ strictly stronger than the trivial $F(n)=o(n)$ recorded in the background; a proof that some fixed prime has an infinite fiber (or a proof that every prime's fiber is finite); a rigorous density result for at least one prime $p$; or a reproducible large-scale computation extending OEIS A039651 that reports records of $F(n)$, the largest fibers observed, and empirical fiber densities, with the code and the range covered. Numerical evidence counts as ADVANCES only when delivered as reproducible code plus the certified range. Deliver the proof and/or the search code with the attained range and measured statistics.