Beatty primes from fractional powers of almost-primes

Let $\alpha >1$ be irrational and of finite type, $\beta \in R$. In this paper, it is proved that for $R⩾13$ and any fixed $c\in (1,c_R)$, there exist infinitely many primes in the intersection of Beatty sequence $B_{\alpha ,\beta }$ and $\lfloor n^c\rfloor$, where $c_R$ is an explicit constant depending on $R$ herein, $n$ is a natural number with at most $R$ prime factors, counted with multiplicity.