Connectedness conjecture for minimizing double bubbles in the three-torus

Let $T^3$ be the flat cubic three-torus, and let an area-minimizing double bubble enclose two prescribed volumes, with the remaining region called the complement.

Connectedness conjecture. An area-minimizing double bubble in $T^3$ has connected regions and complement.

The authors describe this as an intuitive conjecture for which they do not know proofs, although they mention partial results later in the paper.