The q-analog of Rado's theorem for independent q-transversals

Let $M$ be a $q$-matroid on a vector space $V$, with bar nullity function $\overline{n}$, and let $\mathcal X$ be a system of subspaces of $V$. A q-transversal is a subspace obtained by choosing one-dimensional subspaces from the members of $\mathcal X$ as in the paper's definition. The q-analog of Rado's theorem. The system $\mathcal X$ has a $q$-transversal that is independent in $M$ if and only if, for every $J\subseteq I$,

$$\overline{n}(\mathcal X(J))+|J|\leq \overline{n}(V).$$

This is proposed as an avoidance-form $q$-analog of Rado's theorem; the source gives no resolution, so the claim remains open.