The 4-cycle obstruction conjecture for κ-matroidal connectivity functions

Let $E$ be a finite set, and let $u$ be the connectivity function on $\\{a,b,c,d\\}$ defined by

$$\nu(\{a\})=\nu(\{b\})=\nu(\{c\})=\nu(\{d\})=1,$$ $$\nu(\{a,b\})=\nu(\{b,c\})=1,\qquad \nu(\{a,c\})=2,$$

and with all other values determined by symmetry. A connectivity function is unitary when its singleton values are at most one, and it is κ-matroidal when it is obtained by the operation $\mu\downarrow (E-D)$ from a matroidal connectivity function. The 4-cycle obstruction conjecture. Let $\mu$ be a unitary connectivity function on $E$. Then $\mu$ is κ-matroidal if and only if there is no subset $D\subseteq E$ such that $\mu\downarrow (E-D)$ is isomorphic to $\nu$. The conjecture proposes that the connectivity function arising from a 4-cycle is the only obstruction to being κ-matroidal, a property that is minor-closed under the operation defined in the source. Its status is not resolved in the supplied text.