Schneider's support conjecture for mixed Hessian measures

Fix an open convex set $\Omega\subseteq\mathbb{R}^n$, and let $\operatorname{Conv}(\Omega)$ be the set of proper convex functions $f$ on $\mathbb{R}^n$ with $\Omega\subseteq\mathop{\mathrm{dom}} f$. For $f\in\operatorname{Conv}(\Omega)$ and $x\in\Omega$, let $L(f,x)$ be the largest convex subset of $\Omega$ on which $f$ is affine and that has $x$ in its relative interior, and set $\bar L(f,x)=\operatorname{span}\{L(f,x)-x\}$. A point $x\in\Omega$ is $(f_1,\ldots,f_n)$-extreme if

$$\dim\big(\bar L(f_I,x)^\perp\big)\geq |I|$$

for every $I\subseteq[n]$.

Schneider's conjecture. For $f_1,\ldots,f_n\in\operatorname{Conv}(\Omega)$, the mixed Hessian measure satisfies

$$\mathop{\mathrm{supp}} \mathrm{H}_{f_1,\ldots,f_n}=\mathop{\mathrm{cl}}\big\{x\in\Omega:x\text{ is }(f_1,\ldots,f_n)\text{-extreme}\big\}.$$

This is the convex-function analogue of Schneider's conjectural support characterization for mixed area measures. The source presents it as a transcription of that conjecture, and no resolution is supplied.