Characterization conjecture for $L_p$-surface area measures

Let $p\in(0,1)$, and let $\mu$ be a non-trivial Borel measure on $\mathbb S^{n-1}$. A convex body $K\in\mathcal K_0^n$ has $L_p$-surface area measure $\mu$.

Characterization conjecture. The measure $\mu$ is the $L_p$-surface area measure of a convex body $K\in\mathcal K_0^n$ if and only if $\operatorname{supp}\,\mu$ is not a pair of antipodal points.

This conjecture seeks to characterize the measures arising in the $L_p$-Minkowski problem for $p\in(0,1)$. It is proved in the planar case $n=2$ independently by Böröczky and Trinh and by Chen, Li, and Zhu; the status in general dimension is not established here.