Is there a version of the Bernstein von Mises theorem that allows us to perform a Gaussian approximation of the posterior density (assuming such a density with respect to the Lebesgue measure exists)? Most versions of the theorem I am aware of usually establish convergence in distribution or in Total Variation (i.e.
$$d_{TV}\left(\Pi(\cdot | X_{1:n}), \mathcal{N}(\theta^*, (nI_{\theta^*})^{-1})\right) \to 0 \quad \text{a.s. as }n\to \infty$$
Is there a version of the theorem that would yield $L_{\infty}$ or even pointwise convergence of the respective densities in a parametric setting?