Quantitative Erdős–Kac conjecture for irreducible quadratic polynomials

Let $\eta_N\searrow0$ as $N\rightarrow\infty$. For real numbers $\alpha>0$, $\beta$, and $\gamma$ such that $\alpha$ and $\beta$ are not both rational and $\beta^2<4\alpha\gamma$, define

$$X_N=\frac{\omega(\lfloor\alpha n^2+\beta n+\gamma\rfloor)-\log\log N}{\sqrt{\log\log N}},\qquad n\sim U[N].$$

Quantitative Erdős–Kac conjecture for irreducible quadratics. The Kolmogorov distance between $X_N$ and the standard Gaussian $\mathcal{N}(0,1)$ is $O_{\alpha,\beta,\gamma}(\eta_N)$ as $N\rightarrow\infty$, with the implied constant allowed to depend on $\alpha$, $\beta$, and $\gamma$.

The conjecture concerns quantitative convergence for irreducible quadratic polynomials, after the paper's general construction of arbitrarily slow convergence for some reducible generalized polynomials. The linear irreducible case is covered by the Beatty-sequence result, while this quadratic case remains open.