Let $\delta(m,\alpha)$ denote the density of the set of integers which are divisible by some $d\equiv 1\pmod{m}$ with $1<d<\exp(m^\alpha)$. Does there exist some $\beta\in (1,\infty)$ such that

$$\lim_{m\to \infty}\delta(m,\alpha)$$

is $0$ if $\alpha<\beta$ and $1$ if $\alpha>\beta$?