Algebraic matroid basis conjecture for Grassmannians and coherent matching fields

Let $\operatorname{Gr}(k,n)$ be the Grassmannian, and let $M_{\operatorname{Gr}(k,n)}$ be its algebraic matroid. For a coherent matching field $L$, let $M_L$ be the algebraic matroid realised by the linear span of the relevant Gröbner-fan cone. Algebraic matroid basis conjecture. Every basis of the algebraic matroid of the Grassmannian is a basis of $M_L$ for some coherent matching field $L$, i.e.,

$$\mathcal B(M_{\operatorname{Gr}(k,n)})=\bigcup_{L\text{ coherent}}\mathcal B(M_L).$$

The inclusion from each matching-field matroid into the Grassmannian matroid is known; the conjecture asserts the reverse inclusion. It is motivated by computations in all small examples considered, while non-matching-field cones of the tropical Grassmannian show that the claim is not immediate.