Rigidity conjecture for backward mean curvature flows in spheres

Determine whether long-time backward mean curvature flows in spheres exhibit a rigidity property modeled on isoparametric hypersurfaces, as proposed by Liu and Terng; specifically, resolve the conjectural implication relating the relevant asymptotic behavior of such a flow to an isoparametric initial hypersurface.

Background

The paper places its study in the context of backward mean curvature flows (BMCFs) starting from isoparametric hypersurfaces and equifocal hypersurfaces. Liu and Terng established long-time existence for BMCFs starting from isoparametric submanifolds and obtained explicit asymptotic descriptions in the hypersurface case.

The authors state that Liu and Terng proposed a rigidity conjecture for long-time BMCF solutions in spheres modeled on isoparametric hypersurfaces. The quoted passage does not provide the conjecture's detailed mathematical formulation, so the problem is recorded only at the level explicitly stated there.