Erdős Problem #697 — Density threshold for divisibility by restricted congruence divisors

For each $m\in\mathbb N$ and $\alpha\in\mathbb R$, let $\delta(m,\alpha)$ be a density of the set $\{n\in\mathbb N:\exists d,\ d\equiv1\pmod m,\ 1<d<\exp(m^\alpha),\ d\mid n\}$. Is it true that, if $\frac1{\log 2}<\alpha$, then $\lim_{m\to\infty}\delta(m,\alpha)=0$, while if $\alpha<\frac1{\log 2}$, then $\lim_{m\to\infty}\delta(m,\alpha)=1$?