The tropical halfspace representation conjecture for pure tropical polytopes

Let $P$ be a tropical polytope, let $\overline{P}$ be a lift of $P$, and call the lift generic when it satisfies the genericity condition used in the source. Call $P$ pure when all of its maximal cells have the same dimension, and call $P$ generic when it satisfies the source's genericity condition.

Tropical halfspace representation conjecture. If $P$ is pure, then the halfspaces from a generic lift $\overline{P}$ of $P$ map to tropical halfspaces whose intersection is $P$ itself. If, in addition, $P$ is generic, then any lift works.

The preceding proposition proves the assertion under the additional hypothesis that the relevant intersection of tropical halfspaces is pure; the conjecture asserts that this purity conclusion holds in general.