Multivariate and quantitative Erdős-Kac laws for Beatty sequences

The classical Erdős-Kac theorem states that for $n$ chosen uniformly at random from $1, \dots, N$, the random variable $(ω(n) - \log\log N)/\sqrt{\log\log N}$ converges in distribution to the standard Gaussian as $N$ tends to infinity. Banks and Shparlinski showed that this Gaussian convergence holds for any Beatty sequence $\lfloorαn + β\rfloor$ in place of $n$. Continuing in this spirit, Crnčević, Hernández, Rizk, Sereesuchart and Tao considered the joint distribution of $ω(n)$ and $ω(\lfloorα...