Conjecture on Poincaré duality and ring structures for tropical varieties

Let $F$ be a tropical variety. Write $\overline{HH}^\bullet(F)$ for its large cohomology, $\overline{H}_\bullet(F)$ for its large homology, and let

$$D_F:\overline{HH}^\bullet(F)\to\overline{H}_\bullet(F)$$

be the Poincaré duality map. A tropical variety is regularizable in codimension 1 when it has the regularizability property in codimension $1$, and it is irreducible in the usual tropical sense. For a regularizable, irreducible $F$, let $\mathcal{A}$ denote the ring structure on $\overline{H}_\bullet(F)$ induced by the product on $\overline{HH}^\bullet(F)$, and let $\mathcal{B}$ denote the ring structure induced by the ambient space when $F$ is diagonalizable.

Poincaré duality and ring-structure conjecture. The map $D_F$ is surjective for every tropical variety $F$. If $F$ is regularizable in codimension $1$ and irreducible, then its kernel is an ideal, $F$ is diagonalizable, and the induced ring structures $\mathcal{A}$ and $\mathcal{B}$ on $\overline{H}_\bullet(F)$ coincide.

The conjecture packages expected compatibility between large tropical cohomology, large tropical homology, and the ambient intersection product. The source gives no resolution status for these assertions; in particular, the claims remain open here.