Even $L_p$-Minkowski uniqueness conjecture

Let $h$ be the support function of an origin-symmetric convex body in $\mathbb{R}^{n+1}$, let $f$ be a positive even smooth function on $S^n$, and let $\nabla^2h+h\operatorname{Id}$ denote the spherical Hessian matrix.

Even $L_p$-Minkowski uniqueness conjecture. For $p\in[0,1)$, the Monge–Ampère equation

$$h^{1-p}\det(\nabla^2h+h\operatorname{Id})=f$$

has a unique even solution.

This is the uniqueness form of the even $L_p$-Minkowski problem and is equivalent to the even $L_p$ Brunn–Minkowski inequalities. The source describes uniqueness for $0\leq p<1$ as a major unresolved problem.