The tropical halfspace characterization conjecture for pure full-dimensional polytopes

Let $P$ be a tropical polytope. A halfspace description of $P$ is a description as an intersection of tropical halfspaces, and the apex of each halfspace is its distinguished point. The tropical semiring is $\bigl(\mathbb{R},\operatorname{min},+\bigr)$. Call the apices in general position when they satisfy the source's general-position condition.

Tropical halfspace characterization conjecture. A tropical polytope $P$ is pure and full dimensional if and only if it has a halfspace description whose apices are in general position with respect to the tropical semiring $\bigl(\mathbb{R},\operatorname{min},+\bigr)$.

This conjecture proposes an intrinsic halfspace characterization of purity and full dimensionality for tropical polytopes. The supplied text gives no resolution or further evidence.