Strict ML-degree monotonicity under the weak order on matroids

Let $v$ and $w$ be scalings for $\mathbb P^{m-1}\times\mathbb P^{n-1}$, and define

$$\widehat v=[I_m\mid v],\qquad \widehat w=[I_m\mid w].$$

Let $M_v$ and $M_w$ be the linear matroids defined by the columns of $\widehat v$ and $\widehat w$, respectively, and let $\mathcal S(M)$ denote the corresponding matroid stratum. Strict weak-order monotonicity conjecture. If $\widehat v\in\mathcal S(M_v)$, $\widehat w\in\mathcal S(M_w)$, and $M_v<M_w$, then

$$\operatorname{mldeg}(X_{A,v})<\operatorname{mldeg}(X_{A,w}).$$

The claim proposes that ML degree strictly increases along the weak order on matroids; its general validity is open.