Stability of standard planar double bubbles under area-preserving gradient flows

Let a standard planar double bubble evolve by a sufficiently smooth area-preserving gradient flow. Stability conjecture. Standard planar double bubbles are stable under sufficiently smooth area-preserving gradient flows. The paper proves stability for the surface diffusion flow and proposes this statement as a generalization to other area-preserving gradient flows; the conjecture remains open in the stated generality.