Dual equivalence-relation conjecture for q-matroids

Let $M$ be a $q$-matroid, and let $M^*$ denote its dual. The relation of Definition is defined on $M$ and $M^*$.

Dual equivalence-relation conjecture. The relation of Definition is an equivalence relation in at least one of $M$ and $M^*$.

The conjecture is motivated by examples in which the relation is an equivalence relation for some $q$-matroids but does not behave as desired for direct-sum decompositions or under duality. Its resolution is not established in the supplied text.