Minimal mixed volume implies standard-simplex containment

Let $\Sigma_1,\ldots,\Sigma_m$ be an essential collection of integer polyhedra in $\mathbb{R}^m$, and let their mixed volume be $1/m!$, the minimal possible value. Minimal mixed-volume conjecture. There exist an automorphism $A:\mathbb{Z}^m\to\mathbb{Z}^m$ and vectors $a_1,\ldots,a_m\in\mathbb{Z}^m$ such that

$$A(\Sigma_1)+a_1,\ldots,A(\Sigma_m)+a_m$$

are contained in the standard simplex. This is a structural claim about collections attaining the minimum possible mixed volume; the supplied text does not state whether it has been proved or remains open.