Uniqueness conjecture for capillary Gauss curvature solitons

Let $\mathcal{C}_{\theta}$ be the spherical cap and consider the capillary soliton equation referred to as Eq.

for strictly convex capillary hypersurfaces, with contact angle $\theta\in(0,\pi)$. **Capillary soliton uniqueness conjecture.** For $\theta\in(0,\pi)$, the spherical \cap $\mathcal{C}_{\theta}$ is the unique strictly convex solution to Eq.

. The preceding theorem shows convergence of the normalized capillary Gauss curvature flow to a strictly convex capillary soliton. Classifying all such solitons is stated to be an open problem, and the conjecture would identify the spherical cap as the only one.