Left-right invariance criterion for abstract Wiener-group heat kernels

Let $\mathbf H\subset\mathbf G$ be an abstract Wiener group with Lie algebra $\mathbf h$, let $\mathbf H_0$ be the identity component of $\mathbf H$, and let $[\nu_T^{\mathbf H\subset\mathbf G}]$ be the heat-kernel measure class for $T>0$. Write $GL(\mathbf h)_{(\mathcal L_2)}$ for the group of Hilbert-space operators whose deviation from the identity is Hilbert–Schmidt. Left-right invariance conjecture. For $T>0$ and $h\in\mathbf G$, $[\nu_T^{\mathbf H\subset\mathbf G}]$ is left and right $h$-invariant if and only if $h\in\mathbf H_0$ and $\operatorname{Ad}(h)\in GL(\mathbf h)_{(\mathcal L_2)}$. The conjecture is motivated by the Cameron–Martin–Segal theorem and by the relation between translation invariance and conjugation invariance for inversion-invariant heat kernels.