Multi-Bubble Isoperimetric Conjecture in Gaussian space

Let $\mathbb{G}^n$ denote Gaussian space, and let $\Omega^{k+1}[\mathbb{G}^n]$ be a standard simplicial $k$-bubble, meaning a $(k+1)$-partition obtained by translating a centered simplicial partition of $\mathbb{R}^n$. Let $V_{\gamma^n}(\Omega)$ be its Gaussian volume vector, and let $\Delta^{(k)}[1]$ be the admissible region of volume vectors with total mass $1$. Multi-Bubble Isoperimetric Conjecture. For all $2 \leq k \leq n$, a standard simplicial $k$-bubble uniquely minimizes total Gaussian perimeter among all $k$-clusters $\Omega$ on $\mathbb{G}^n$ of prescribed Gaussian volume $V_{\gamma^n}(\Omega) = v \in \operatorname{int} \Delta^{(k)}[1]$. This is the Gaussian analogue of the Euclidean, spherical, and hyperbolic multi-bubble problem; the two-dimensional cases are established, whereas the higher-dimensional cases remain open.