Uniqueness conjecture for power capillary Gauss curvature solitons

Let $\mathcal{C}_{\theta}$ be the spherical cap, let $\alpha>\frac{1}{n+2}$ and let $\theta\in(0,\pi)$. Suppose that $h$ is a positive, strictly convex function on $\mathcal{C}_{\theta}$, with $\nabla^2h$ its covariant Hessian and $\sigma$ the standard metric. Power capillary soliton uniqueness conjecture. If

$$\left\{ \begin{array}{llll} h\det^\alpha(\nabla^2 h+h\sigma)=\ell & \text{ in } \mathcal{C}_\theta,\\ \nabla_\mu h=\cot\theta\,h & \text{ on } \partial\mathcal{C}_\theta, \end{array} \right.$$

then $h=\ell$. The source says that classification of the corresponding capillary solitons remains open and presents this boundary-value formulation as an equivalent conjecture; it is intended to characterize the unit spherical cap after normalizing the Lagrange multiplier to $1$.