Jackson–Tanigawa's hyperconnectivity maximality conjecture

Let $n\geq 2d+2$ and let $E$ be the edge set of the complete graph $K_n$. For a graph family $\mathcal{X}$, an $\mathcal{X}$-matroid is a matroid on $E$ in which every graph in $\mathcal{X}$ is a circuit; $\operatorname{val}_{\mathcal{X}}$ denotes the associated upper-bound function on subsets of $E$. The $d$-hyperconnectivity matroid $\mathcal{H}_n^d$ is the generic $d$-hyperconnectivity matroid on $n$ vertices.

Jackson–Tanigawa's hyperconnectivity conjecture. For $d\geq 1$ and $\mathcal{X}=\{K_{d+2},K_{d+1,d+1}\}$, $\mathcal{H}_n^d$ is the unique maximal $\mathcal{X}$-matroid, and $\operatorname{val}_{\mathcal{X}}$ is its rank function.

The conjecture is true for $d=1$; it remains open for $\mathcal{H}_n^2$, while the paper disproves it for $d\geq 3$ by constructing maximal $\mathcal{X}$-matroids distinct from $\mathcal{H}_n^d$.