Beatty primes from fractional powers of almost-primes

Let $\alpha>1$ be irrational and of finite type, $\beta\in\mathbb{R}$. In this paper, it is proved that for $R\geqslant13$ any fixed $c\in(1,c_R)$, there exist infinitely many primes in the intersection Beatty sequence $\mathcal{B}_{\alpha,\beta}$ $\lfloor n^c\rfloor$, where $c_R$ an explicit constant depending on $R$ herein, $n$ a natural number with at most prime factors, counted multiplicity.