Uniqueness conjecture for constant Gaussian-Minkowski data in higher dimensions

Let $n\geq 3$, and let $h$ be a nonnegative solution on the unit sphere $S^{n-1}$ of

$$\frac{1}{(\sqrt{2\pi})^n}e^{-\frac{|\nabla h|^2+h^2}{2}}\det(\nabla^2h+hI)=c>0.$$

Higher-dimensional constant-data uniqueness conjecture. Then $h$ must be a constant solution.

The equation is the higher-dimensional analogue of the Gaussian-Minkowski equation studied in the paper. The preceding discussion explains that extending the existence results to higher dimensions depends on establishing uniqueness for constant $f$; the status of this assertion is not resolved in the supplied text.