Algebraic-geometric genericity conjecture for the combinatorial derived matroid

Fix a field $\mathbb{F}$ with algebraic closure $\overline{\mathbb{F}}$. Let $M$ be representable over $\mathbb{F}$, with $|E(M)|=n$ and $r(M)=k$, and let $V(M)\subseteq\overline{\mathbb{F}}^{k\times n}$ be the variety of matrices representing $M$. For each $\overline{\mathbb{F}}$-representable matroid $N$ on the circuit set $\mathcal{C}=\mathcal{C}(M)$, let $V_N(M)=\{R\in V(M):\delta_{OW}(R)=N\}$. Algebraic-geometric genericity conjecture. For every matroid $N\neq\delta(M)$ on $\mathcal{C}$, if $V_N(M)$ is non-empty, then it has positive codimension in $V(M)$. This would imply that representations yielding the combinatorial derived matroid form a full-dimensional locus, and over growing finite extensions would make them asymptotically almost all representations.