Categorical genericity conjecture for the combinatorial derived matroid

Let $M$ be a representable matroid with set of circuits $\mathcal{C}$. For matroids on a fixed ground set, use the weak order defined by $M_1\geq M_2$ when every dependent set of $M_1$ is dependent in $M_2$. Categorical genericity conjecture. Then

$$\delta M\geq\delta_{OW}(R)$$

for every representation $R$ of $M$, where the order is the weak order on matroids with ground set $\mathcal{C}$. This claims that the combinatorial derived matroid is the generic, maximal derived matroid in the categorical weak-order sense.