Bernstein's conjecture on algebraic matroids of Hadamard products

Let $L_1, \ldots, L_d \subset \mathbb{C}^m$ be linear spaces not contained in any coordinate hyperplane, and let $r_1, \ldots, r_d$ be the rank functions of the corresponding linear matroids. Bernstein's conjecture. The algebraic matroid of the Hadamard product $L_1 * \cdots * L_d$ is the matroid induced by the monotone submodular function $r_1 + \cdots + r_d - (d-1)$. This conjectures the natural $d$-factor generalisation of Bernstein's theorem for two Hadamard factors; the two-factor case is known, while the general case is presented here as an open problem.