Dual defectivity and connectedness of the matroid

Let $L$ be a linear space with associated matroid $M(L)$, and let $L^{-1}$ be its reciprocal linear space. Write $abla(L^{-1})$ for the dual variety of $L^{-1}$.

Dual-defectivity conjecture. The reciprocal linear space $L^{-1}$ is dual defective, meaning

$$\dim \nabla(L^{-1})<n-1,$$

if and only if the matroid $M(L)$ is not connected.

This conjecture characterizes dual defectivity in matroidal terms. It was first proposed by Clara Briand; if true, it identifies the connected flats as the relevant building-set components in the Newton-polytope description of the principal matroid determinant.