Schneider's support conjecture for mixed area measures

Let $C_1,\ldots,C_{n-1}$ be convex bodies in $\mathbb{R}^n$. For $I\subseteq[n-1]$, write $C_I=\sum_{i\in I}C_i$, and let $T(C_I,u)$ denote the touching cone of $C_I$ with normal direction $u$. A point $u\in S^{n-1}$ is $(C_1,\ldots,C_{n-1})$-extreme if

$$\dim\big(T(C_I,u)^\perp\big)\geq |I|$$

for every $I\subseteq[n-1]$.

Schneider's conjecture. The support of the mixed area measure satisfies

$$\mathop{\mathrm{supp}} \mathrm{S}_{C_1,\ldots,C_{n-1}}=\mathop{\mathrm{cl}}\big\{u\in S^{n-1}:u\text{ is }(C_1,\ldots,C_{n-1})\text{-extreme}\big\}.$$

For polytopes, this follows from the characterization of positivity of mixed volumes, while the assertion for general convex bodies gives the conjectural extension of the polytope description and remains unresolved in general.