Hanging-elements conjecture for the combinatorial derived matroid

Let $M$ be a matroid with set of circuits $\mathcal{C}$. For $S\subseteq\mathcal{C}$, let $\operatorname{supp}(S)$ be the union of the circuits in $S$, and let $\mathcal{A}$ be the collection of dependent sets defining the combinatorial derived matroid $\delta M$. Hanging-elements conjecture. If $S\notin\mathcal{A}$ and $C\in\mathcal{C}$ has an element outside $\operatorname{supp}(S)$, then

$$S\cup\{C\}\notin\mathcal{A}.$$

The statement is proposed as an analogue of a known result for the Oxley–Wang derived matroid, but the paper explains that the analogous claim for $\mathcal{A}_0$ is insufficient and that new ideas are needed.