Are there $\gg\phi(d)$ residues $a$ with least prime $p(a,d)>(1+c)\phi(d)\log d$? (Erdős #971)

For $a$ coprime to $d$, let $p(a,d)$ denote the least prime congruent to $a\pmod d$. Does there exist a constant $c>0$ such that, for all large $d$, $$p(a,d)>(1+c)\phi(d)\log d$$ holds for $\gg\phi(d)$ many residues $a$ (coprime to $d$)? Here $\phi$ is Euler's totient, and $\phi(d)\log d$ is the average order of $p(a,d)$ over the admissible residues $a$.

Acceptance. FULLY RESOLVES: a proof (machine-checkable preferred, else a complete written proof) either establishing the existence of a constant $c>0$ for which $p(a,d)>(1+c)\phi(d)\log d$ holds for $\gg\phi(d)$ residues for all large $d$, or disproving it. ADVANCES (proof required): (a) prove the statement for a set of moduli $d$ of positive density, strictly improving on Erdős's 'infinite sequence of $d$'; (b) establish a quantitative lower bound tending to a positive proportion for the number of residues $a$ with $p(a,d)>(1+c)\phi(d)\log d$ for a specified class of $d$; or (c) a rigorous conditional result (e.g. under GRH), clearly flagged as such. Deliver the proof or formalization; numerical distribution tables may support but do not by themselves constitute an advance on this proof-shaped question.