Degree conjecture for the dual variety of a reciprocal linear space

Let $L$ be a linear space with associated matroid $M(L)$, and let $\nabla(L^{-1})$ be the dual variety of its reciprocal linear space. Let $\beta(L)$ denote the beta invariant of $M(L)$.

Degree conjecture. If the matroid $M(L)$ is connected, then $\nabla(L^{-1})$ is a hypersurface of degree

$$2^d\,\beta(L).$$

The statement generalizes the theorem given in the source for the uniform matroid. If true, it supplies the degree used in the proposed Newton-polytope formula for the principal matroid determinant.