Classification conjecture for stable capillary hypersurfaces in a Euclidean ball

Let $M^n$, with $n\geq 3$, be a capillary hypersurface in the unit Euclidean ball $B^{n+1}$. Classification conjecture. The only stable such hypersurfaces are a totally geodesic hyperplane or a spherical cap.

The conjecture proposes a classification of stable capillary hypersurfaces, extending the known stability or instability of Delaunay-type examples. Its status is not resolved in the supplied source.