Wagstaff's asymptotic 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. Wagstaff's conjecture. As $n\to\infty$, for most $n$,

$$P(n)\sim\phi(n)\log(n)\log(\phi(n)).$$

This is the central heuristic prediction developed from Wagstaff's probabilistic model; the paper provides numerical evidence but no proof of the asserted asymptotic.