Erdős–Kac conjecture for integer parts of polynomials with an irrational coefficient

Let $f$ be a polynomial with at least one irrational non-constant coefficient. For $n\sim U[N]$, define

$$X_N=\frac{\omega(\lfloor f(n)\rfloor)-\log\log N}{\sqrt{\log\log N}}.$$

Erdős–Kac conjecture for integer parts of polynomials. The random variable $X_N$ converges in distribution to the standard Gaussian $\mathcal{N}(0,1)$ as $N\rightarrow\infty$.

This would generalise the Erdős–Kac laws established in the paper for Beatty sequences and integer polynomials; the corresponding result for general polynomials with an irrational non-constant coefficient remains open.