The tropical discriminant is a union of cones in the secondary fan

Let $A$ be a point configuration, let $X_A^*$ be its $A$-discriminant variety, let $\tau(X_A^*)$ be the tropical discriminant, and let $\Sigma(A)$ be the secondary fan parametrizing the regular polyhedral subdivisions of $A$. For $w\in\tau(X_A^*)$, denote by $\Pi_w$ the regular subdivision specified by $w$.

Tropical discriminant conjecture. For any point configuration $A$, the tropical discriminant $\tau(X_A^*)$ is a union of cones in the secondary fan $\Sigma(A)$.

In the non-defective case, the $A$-discriminant divides the principal $A$-determinant, so the tropical discriminant is a subfan of the secondary fan. The conjecture asserts that membership in the tropical discriminant depends only on the regular subdivision determined by $w$, even in the defective case.