Submodularity criterion for a unique maximal matroid

Let $\mathcal X$ be a family of subsets of a finite set $E$, and let $\operatorname{val}_{\mathcal X}$ be the set function defined from proper $\mathcal X$-sequences. Suppose there is at least one $\mathcal X$-matroid on $E$. Submodularity criterion. The poset of all $\mathcal X$-matroids on $E$ has a unique maximal element if and only if $\operatorname{val}_{\mathcal X}$ is a submodular set function on $E$. More generally, without assuming that an $\mathcal X$-matroid exists, the poset of all $\mathcal X$-cyclic matroids on $E$ should have a unique maximal element if and only if $\operatorname{val}_{\mathcal X}$ is submodular. The forward implication is supplied by the preceding lemma; the converse is the conjectural part, while the more general cyclic formulation is also proposed.