Finite-obstruction conjecture for κ-matroidal connectivity functions

Let $E$ be a finite set, let $\mu$ be a unitary connectivity function on $E$, and let $\mu\downarrow X$ denote the connectivity-function minor obtained by the operation defined in the source. Finite-obstruction conjecture. There is a finite set $S$ of connectivity functions such that, if $\mu$ is a unitary connectivity function on $E$, then $\mu$ is κ-matroidal if and only if, for every $X\subseteq E$, the connectivity function $\mu\downarrow X$ is not isomorphic to a member of $S$. This is presented as a weaker version of the preceding 4-cycle obstruction conjecture. The supplied text does not resolve whether such a finite obstruction set exists.