Adjoint conjecture for the combinatorial derived matroid

Let $M$ be a matroid with ground-set size $n$ and rank $k$, and let $M^*$ denote its dual. Assume that $M^*$ has an adjoint. Adjoint conjecture. Then $\delta M$ is isomorphic to an adjoint of $M^*$. In particular,

$$\operatorname{rank}(\delta M)=n-k.$$

The conjecture would identify the combinatorial derived matroid with an adjoint whenever the relevant adjoint exists, and would settle the equality case of the general upper bound on its rank.