Higher Moments for Polynomial Chowla

Let $\lambda(n)$ be the Liouville function. We study the distribution of $$\frac{1}{x^{1/2}}\sum_{x\leq n\leq 2x}\lambda(f(n))$$ over random polynomials $f$ of fixed degree $d$ and coefficients bounded in magnitude by $H$. In particular we prove that the first $d+1$ moments are Gaussian.