Bernstein–von Mises conjecture for the kernel variance in normal location mixtures

Let $X_1,X_2,\ldots$ be an independent and identically distributed sample from $P_0=P_{\sigma_0,F_0}$ in the semiparametric normal location mixture model parametrized by the model in the source. Let $\Sigma=[\sigma_-,\sigma_+]\subset(0,\infty)$ have a thick prior, and let $\mathscr D[0,1]$ carry a Dirichlet prior $D_{\alpha}$ with finite base measure $\alpha$ dominating Lebesgue measure on $[0,1]$. Assume that the efficient Fisher information at $P_0$ is nonsingular. Bernstein–von Mises conjecture. The marginal posterior for the kernel variance $\sigma$ should satisfy

$$\sup_{B\in\mathscr B(\Sigma)}\left|\Pi\left(\sqrt n(\sigma-\sigma_0)\in B\mid X_1,\ldots,X_n\right)-N_{\widetilde{\Delta}_n,\widetilde{I}_{\sigma_0,F_0}^{-1}}(B)\right|\mathrel{\longrightarrow^{P_0}}0.$$

This predicts a Bernstein–von Mises limit for the finite-dimensional kernel-variance parameter despite the infinite-dimensional mixing distribution. The statement concerns semiparametric posterior asymptotic normality under the stated prior and information assumptions.