Uniqueness conjecture for the four-region sandwich cluster

Let $\mathcal C(1,1,1,1)$ denote the stationary planar clusters with four regions of area $1$, and let a sandwich-topology cluster be one with the topology described in the source. Sandwich uniqueness conjecture. Up to isometries, there is a unique stationary cluster $\boldsymbol E\in\mathcal C(1,1,1,1)$ with the sandwich topology. In such a cluster, the regions $E_1$ and $E_3$ are isometric to $E_2$ and $E_4$, respectively. The paper proves that equal-area minimizing clusters have sandwich topology, but this stronger uniqueness and symmetry statement is presented as conjectural.