Algorithm for identifying minimal X-matroids

Find an algorithm to identify the set of minimal \(\mathcal{X}\)-matroids for an arbitrary family \(\mathcal{X}\subseteq\binom{[d]}{3}\).

Background

The paper relates its algorithm for minimal matroids of point-line configurations to $\mathcal{X}$-matroids, where every member of a prescribed family $\mathcal{X}$ is required to be a circuit. The authors explain that a slightly modified version of their algorithm can produce all minimal $\mathcal{X}$-matroids of rank at most three, subject to preventing identifications of points lying in a common member of $\mathcal{X}$. They nevertheless explicitly ask for an algorithm addressing the general identification problem for minimal $\mathcal{X}$-matroids.