Smallest prime $\equiv 1\ (\mathrm{mod}\ n)$ versus smallest $m$ with $n\mid\phi(m)$ (Erdős #456)

For a positive integer $n$, let $p_n$ be the smallest prime $\equiv 1\pmod{n}$, and let $m_n$ be the smallest positive integer with $n\mid\phi(m_n)$, where $\phi$ is Euler's totient function. Since $\phi(p_n)=p_n-1$ is divisible by $n$, one always has $m_n\leq p_n$. Three questions of Erdős: (i) Is $m_n<p_n$ for almost all $n$? (ii) Does $p_n/m_n\to\infty$ for almost all $n$? (iii) Are there infinitely many primes $p$ such that $p-1$ is the only $n$ for which $m_n=p$?

Acceptance. FULLY RESOLVES (proof-shaped): a complete proof — machine-checkable (Lean/Coq) preferred, else fully written — settling any one of the three questions: that $\{n:m_n<p_n\}$ has density $1$ (or that it fails on a positive-density set); that $p_n/m_n\to\infty$ on a density-$1$ set (or its negation); or that infinitely many (or only finitely many) primes $p$ have $p-1$ as the unique $n$ with $m_n=p$. ADVANCES: an unconditional density lower bound for $\{n:m_n<p_n\}$ strictly improving whatever explicit density is recorded in the background; a proof of (i), (ii) or (iii) under a clearly stated hypothesis (e.g. GRH); or a reproducible computation of $p_n$ and $m_n$ over an extended range giving certified counts of the exceptional sets and, for (iii), an extended table of the primes $p$ for which $p-1$ is the unique $n$ with $m_n=p$ — with the program and a certificate. Deliver the proof, or the search code plus certified statistics.