Fiori's explicit two-sided 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. Fiori's explicit two-sided conjecture. For all $n>3$,

$$P(n)<3\phi(n)\log(n)\log(\phi(n)),$$

and for all $n>570$,

$$P(n)>0.5\phi(n)\log(n)\log(\phi(n)).$$

This combines the paper's proposed explicit upper and lower bounds and is supported by the computations and outlier heuristics described there; no proof is supplied.