Adjoints and the combinatorial derived matroid of the dual

Let $M$ be a matroid with circuit set $\mathcal{C}$, and suppose that an adjoint of $M$ exists. Let $\delta M^*$ denote the combinatorial derived matroid of the dual matroid. Dual-adjoint conjecture. One adjoint of $M$ is equal to $\delta M^*$, and every adjoint of $M$ is less than $\delta M^*$ in the weak order on matroids with ground set $\mathcal{C}$. This conjecture seeks to relate the closure-based adjoint construction to the iterative dependent-set definition of the combinatorial derived matroid; the paper states that the lack of a satisfactory closure description leaves the comparison unresolved.