The face-structure conjectures for tropical polytopes

Let $P$ be a tropical polytope, and let $F$ and $G$ be faces under the source's lift-based definition: choose a lift $\overline{P}$, take the unions $\overline{F}$ and $\overline{G}$ of all faces of the same dimension mapping into $F$ and $G$, and define their intersection as the degree-map image of $\overline{F}\cap\overline{G}$. Let a $k$-face mean a face of dimension $k$, let an extreme set have the source's meaning, and let a cell complex and its homology have their usual topological meanings.

Face-structure conjectures for tropical polytopes. The following assertions hold: (1) $k$-faces of tropical polytopes are extreme sets; (2) the topological boundary of a $k$-face is a union of $(k-1)$-faces, so the faces form a cell complex; (3) the homology of this cell complex is that of a sphere; (4) the intersection of two faces is well-defined, independent of the lift, and is a contractible union of faces; (5) $k$-faces of tropical polytopes are always contractible; and (6) the faces of a tropical polytope do not depend on the provided vertex set, so using a different vertex set to form $P$ and its lifts gives the same $k$-faces for every $k$.

These conjectures are motivated by the model and other examples, and are presented as improvements on Joswig's face notions. The supplied text gives no resolution or partial results for the individual assertions.