Characterization of uniqueness through submodularity of the refined valuation

Prove or disprove that, whenever the set of \(\mathcal{X}\)-matroids is nonempty, the recursively defined function \(v_{\mathcal{X}}\) is submodular if and only if there exists a unique minimal \(\mathcal{X}\)-matroid.

Background

Jackson and Tanigawa established that submodularity of their valuation function $\operatorname{val}_{\mathcal{X}}$, together with nonemptiness of the set of $\mathcal{X}$-matroids, is sufficient for the existence of a unique minimal $\mathcal{X}$-matroid whose rank function is $\operatorname{val}_{\mathcal{X}}$. The paper gives a counterexample showing that the converse of that earlier formulation is false, then introduces a recursively refined function $v_{\mathcal{X}}$ intended to provide a sharper upper bound on ranks of $\mathcal{X}$-matroids. The authors prove the forward implication for this refined function and explicitly conjecture the converse, yielding the stated equivalence.