Gaussian multi-bubble conjecture

Let $me3$, and let $a_1,\dots,a_m>0$ satisfy

$$\sum_{i=1}^m a_i=1.$$

Let $\Omega_1,\dots,\Omega_m\subseteq\mathbb{R}^{n+1}$ be a minimizing partition for the Gaussian multi-bubble problem: the sets cover $\mathbb{R}^{n+1}$, satisfy $\gamma_{n+1}(\Omega_i)=a_i$, and minimize

$$\sum_{1\le i<j\le m}\int_{(\partial\Omega_i)\cap(\partial\Omega_j)}\gamma_n(x)\,dx.$$

Assume that $m-1\le n+1$. Let $z_1,\dots,z_m\in\mathbb{R}^{n+1}$ be the vertices of a regular simplex centered at the origin. Gaussian Multi-Bubble Conjecture. There exists $w\in\mathbb{R}^{n+1}$ such that, for every $1\le i\le m$,

$$\Omega_i=w+\left\{x\in\mathbb{R}^{n+1}:\langle x,z_i\rangle=\max_{1\le j\le m}\langle x,z_j\rangle\right\},$$

with $\gamma_{n+1}(\Omega_i)=a_i$. The conjecture predicts that regular-simplex partitions, translated to realize the prescribed Gaussian volumes, minimize total Gaussian interface area. Its relationship to the standard simplex and plurality-is-stablest conjectures is discussed in the paper; the stated general case is not resolved here.