Stable capillary hypersurfaces in a wedge

Let Σ be a compact immersed stable capillary hypersurface in a wedge bounded by two hyperplanes in Rⁿ⁺¹. Suppose that Σ meets those two hyperplanes in constant contact angles ≥π/2 and is disjoint from the edge of the wedge, and suppose that ∂Σ consists of two smooth components with one in each hyperplane of the wedge. It is proved that if ∂Σ is embedded for n = 2, or if each component of ∂Σ is convex for n ≥ 3, then Σ is part of the sphere. The same is true for Σ in the half-space of Rⁿ⁺¹ with connected boundary ∂Σ.