Moment computation conjecture for the optimal estimator

Let $\mathcal{P}_{\bm{t}}$ and $\bm{V}_s$ be as in Proposition. For a coordinate $k$, define the moment

$$\mu_k:=\int_{\mathcal{P}_{\bm{t}}} x_k\,d\varrho_{\mathcal{P}}=\operatorname{vol}(\mathcal{P}_{\bm{t}})\hat{x}_k.$$

Let $\Delta$, $\boldsymbol{\lambda}$, and the inverse Laplace operators $\mathcal{L}_{\lambda_m}^{-1}$ be as in the preceding construction. Moment computation conjecture. Provided that the integrals on the right-hand side exist,

$$\mu_k=\left(\mathop{\circ}_{m}\mathcal{L}_{\lambda_m}^{-1}\right)\left(\int_{\Delta}x_k\exp\bigl(-\langle\bm{V}_s\boldsymbol{\lambda},\bm{x}\rangle\bigr)\,d\bm{x}\right).$$

This conjecture supplies the final moment-computation step for the optimal estimator. The source reports that it was numerically verified, and the parser status indicates that its numerical correctness was checked extensively against integration by simplicial decomposition, although a rigorous proof is not supplied here.