The soap bubble conjecture for planar clusters

Let $N$ be a positive integer, let $\boldsymbol a\in\mathbb{R}^N_+$ be a vector of prescribed positive areas, and let $\mathcal M(\boldsymbol a)$ denote the set of perimeter-minimizing clusters with those areas. Soap bubble conjecture. For every $\boldsymbol a\in\mathbb{R}^N_+$, every $\boldsymbol E\in\mathcal M(\boldsymbol a)$ is connected. This is the general connectedness question for minimizing planar clusters; the paper proves the claim for four equal-area regions, while the general case remains open.