The partition characterization of minimal tropical halfspaces

Let ${\mathcal P}$ be a tropical polytope in the generic case. For a vertex $v$ of ${\mathcal P}$, let the grid label of $v$ have shaded boxes in its columns, and let $C_i$ denote the sector associated with index $i$. A tropical halfspace has the form $v+\bigcup_{i\in A}C_i$ for a nonempty proper subset $A$ of the sector indices.

Partition characterization. Every minimal halfspace with respect to ${\mathcal P}$ has the form

$$v+\bigcup_{i\in A}C_i,$$

where $v$ is a vertex of ${\mathcal P}$ and the shaded boxes in the columns indexed by $A$ form a partition of $[r]$.

This would refine the generally redundant halfspace description of tropical polytopes by characterizing minimal halfspaces through the grid labels of their vertices. The supplied text does not give a proof or indicate whether the assertion has been resolved.