Density convergence in the Breuer-Major theorem for Gaussian stationary sequences

Consider a Gaussian stationary sequence with unit variance $X={X_k;k\in {$\mathbb{N}$}\cup{0}}$. Assume that the central limit theorem holds for a weighted sum of the form $V_n=n^{-1\mathrm{/}2}{\sum }_{k=0}^{n-1}f(X_k)$, where $f$ designates a finite sum of Hermite polynomials. Then we prove that the uniform convergence of the density of $V_n$ towards the standard Gaussian density also holds true, under a mild additional assumption involving the causal representation of $X$.