Bianchi–Böröczky–Colesanti–Yang existence conjecture for the $L_p$-surface area measure

Let $0<p<1$ and let $\mu$ be a finite Borel measure on the unit sphere $\mathbb{S}^{n-1}$ that is not concentrated in a pair of antipodal points. For a convex body $K$ in $\mathbb{R}^n$, write $S_p(K,\cdot)$ for its $L_p$-surface area measure.

Bianchi–Böröczky–Colesanti–Yang conjecture. There exists a convex body $K$ in $\mathbb{R}^n$ such that

$$S_p(K,\cdot)=\mu.$$

The conjecture gives an existence criterion for the $L_p$-Minkowski problem in the range $0<p<1$. The supplied status evidence says that the assertion has been resolved; in the plane it was confirmed independently by Trinh and by Chen, Li, and Zhu.