Fiori's lower-bound conjecture for the least prime in an arithmetic progression

Let $n$ be a positive integer, let $a<n$ satisfy $\operatorname{gcd}(a,n)=1$, and define

$$P(n,a)=\min\{p\mid p\text{ is prime and }p\equiv a\pmod n\},$$ $$P(n)=\max_{(a,n)=1}P(n,a),$$

where $\phi$ denotes Euler's totient function. For $0<\epsilon<1$, define

$$F_\epsilon=\{n\mid P(n)<(1-\epsilon)\phi(n)\log(n)\log(\phi(n))\},$$ $$F_\epsilon(x,y)=\{n\mid x<n<y,\ P(n)<(1-\epsilon)\phi(n)\log(n)\log(\phi(n))\}.$$

Fiori's lower-bound conjecture. For $0<\epsilon<1$, as $y\to\infty$,

$$F_\epsilon(x,y)\ll\int_x^y e^{-n^\epsilon/(\log(n)\log\log(n))}\,dn,$$

and

$$\liminf_{n\to\infty}\frac{P(n)}{\phi(n)\log(n)\log(\phi(n))}=1.$$

In particular, $F_\epsilon$ is finite for every $0<\epsilon<1$. This complements the predicted upper behavior and is supported by the paper's numerical tables, but remains unproved in the supplied text.