Bounded gaps between primes of special form

Let $0<\alpha$, $\sigma<1$ be arbitrary fixed constants, let $q_1<q_2<\dots<q_n<q_{n+1}<\dots$ be the set of primes satisfying the condition $\{q_n^\alpha\}<\sigma$ and indexed in ascending order, and let $m\ge1$ be any fixed integer. Using an analogue of the Bombieri–Vinogradov theorem for the above set of primes, upper bounds are obtained for the constants $c(m)$ such that the inequality $q_{n+m}-q_n\le c(m)$ holds for infinitely many $n$.