Erdős's totient-fiber conjecture: $\varphi(m)=n$ has more than $n^{1-\varepsilon}$ solutions for infinitely many $n$

Theorem 1.1: for every $\varepsilon>0$ there are infinitely many $n$ with $g(n)>n^{1-\varepsilon}$, which is optimal in the exponent because $g(n)\ll_\eta n^{1+\eta}$. The input is Theorem 1.2: for fixed $0<\delta<1/4$, at least $x^{1-o(1)}$ primes $p\in(2x,5x]$ have $P^+(p-1)\le x^\delta$; in particular there are infinitely many primes with $P^+(p-1)\le p^\varepsilon$ for every $\varepsilon>0$. It does not give a positive proportion of such primes (that stronger statement is in the Poisson-Dirichlet companion), and the $o(1)$ is not made explicit.

Theorem 1.1: for every $\varepsilon>0$ there are infinitely many $n$ with $g(n)>n^{1-\varepsilon}$, which is optimal in the exponent because $g(n)\ll_\eta n^{1+\eta}$. The input is Theorem 1.2: for fixed $0<\delta<1/4$, at least $x^{1-o(1)}$ primes $p\in(2x,5x]$ have $P^+(p-1)\le x^\delta$; in particular there are infinitely many primes with $P^+(p-1)\le p^\varepsilon$ for every $\varepsilon>0$. It does not give a positive proportion of such primes (that stronger statement is in the Poisson-Dirichlet companion), and the $o(1)$ is not made explicit.