The converse characterization of matroidal cycles under integer linear equivalence

Let $M$ and $M'$ be non-isomorphic matroids, and let $Z_M$ and $Z_{M'}$ denote their matroidal cycles in their respective ambient vector spaces. Suppose there exists an invertible map

$$\phi: Z_M \to Z_{M'}$$

which is the restriction of an invertible integer linear map between the ambient spaces. Parallel-connection conjecture. Then $M$ and $M'$ are parallel connections. This proposes a converse to the preceding result that parallel connections yield matroidal cycles related by invertible integer linear maps; it is open here because no resolution is supplied.