No periodic three-volume foam conjecture for the cubic three-torus

Let $T^3$ be the standard cubic three-torus, and consider least-area divisions into three volumes. A division is said to lift to a foam in $\mathbb{R}^3$ when its lift under the universal covering has the structure of a foam.

No periodic three-volume foam conjecture. There are no least-area divisions of $T^3$ into three volumes that lift to a foam in $\mathbb{R}^3$.

The conjecture is motivated by the regularity of triple-bubble interfaces: singular curves meet in fours, making a lifted double-bubble-type division incompatible with an ordinary periodic foam. The source presents the assertion as conjectural.