Conjecture on phase portraits of minimizing double bubbles on flat tori

Let $A_0\geq A_2\geq A_1$ be areas, and let a flat torus have area $A_0+A_1+A_2$. A non-tiling minimizing double bubble is a perimeter-minimizing double bubble enclosing areas $A_1$ and $A_2$ that does not tile the torus. Phase-portrait conjecture. The non-tiling minimizing double bubbles enclosing $A_1,A_2$ on this torus are the same as the minimizing double bubbles enclosing $A_1,A_2$ on the infinite cylinder. Consequently, the phase portrait of any flat two-torus can be obtained from the phase portrait of any flat two-torus of larger area, or from the phase portrait of the infinite cylinder; in particular, tori of equal areas have identical phase portraits. This conjecture asserts a reduction from flat-torus phase portraits to the infinite-cylinder problem. The source presents it as a conjecture and gives no resolution.