Dowling geometry Kinser-inequality conjecture

Let $G$ be an abelian group, and let a Dowling geometry be the matroid constructed from the corresponding labelled graph whose circuits are the positive cycles and minimal connected subgraphs containing two negative cycles. A Kinser inequality is the rank inequality indexed by an integer $n\geq 4$. Dowling geometry Kinser-inequality conjecture. A Dowling geometry satisfies every Kinser inequality.

The source notes that this is open when $G$ is finite and non-cyclic; finite subgroups of the multiplicative group of a field are cyclic, so the representable cases do not settle the non-cyclic case.