Vanishing conjecture for the cubic generators of dual matroid toric ideals

Let $L$ be a linear subspace of dimension $r$ whose matroid $\mathrm{M}(L)$ is connected. Set $\mathcal E_L=\mathcal S_L\oplus\mathcal Q_L$. Vanishing conjecture for dual matroid cubics. The vector bundle

$$\bigwedge^{r}\mathcal E_L^\vee\otimes\bigwedge^{r}\mathcal E_L^\vee\otimes\det\mathcal Q_L$$

has vanishing $H^1$. The prediction is motivated by the absence of cubic minimal generators for the toric ideal of the dual matroid, but the source does not specify its resolution status.