Let be a matroid on an -element set that is a disjoint union of independent sets of size . Assume that there exists another matroid on the same ground set with the following properties:

(1) is strongly base orderable.

(2) for all , where is the rank function of .

(3) All circuits of satisfying remain dependent in .

Then there is an grid whose th row comprises and whose columns are independent in .

That all strongly base-orderable matroids satisfy Rota’s Basis Conjecture follows at once if we take since condition (2) is automatic by pigeonhole. However, there is a lot more that we might extract out of Lemma 6. Wild asks whether a suitable *always* exists. As Wild recognizes, this is probably too optimistic, but he doesn’t have a counterexample. Maybe a suitable always exists for graphic matroids?