The chirotopal tropical Grassmannian conjecture for rank three

Fix a realizable chirotope $\chi\in\{-1,1\}^{\binom{n}{3}}$. Let $\operatorname{Dr}^\chi_{3,n}$ be the chirotopal Dressian, consisting of the $\chi$-tropical Plücker vectors, and let $\operatorname{Trop}^\chi G(3,n)$ be the realizable $\chi$-tropical Grassmannian. Work modulo the lineality space consisting of vectors with coordinates

$$\pi_J=\sum_{j\in J}x_j$$

for $x\in\mathbb R^n$. Chirotopal tropical Grassmannian conjecture. Each $\operatorname{Dr}^\chi_{3,n}$ is a pure $(3-1)(n-3-1)$-dimensional polyhedral fan, and

$$\operatorname{Trop}^\chi G(3,n)=\operatorname{Dr}^\chi_{3,n}.$$

Moreover, for any fixed maximal cone in $\operatorname{Trop}G(3,n)$, either the cone is contained in no chirotopal tropical Grassmannian, or it is contained in exactly $2^{(3-1)(n-3-1)}$ of them. The conjecture generalizes the known characterization of the positive tropical Grassmannian by the three-term tropical Plücker relations; it asserts realizability for all rank-three realizable chirotopes and includes the stated purity and counting properties.