Bérczi–Schwarcz–Yamaguchi coverability conjecture for matroids

Let ${\cal M}=(E,\mathcal{I})$ be a matroid. It is $k$-coverable if its ground set $E$ can be covered by at most $k$ independent sets from $\mathcal{I}$. A partition matroid on the same ground set $E$ is likewise $k$-coverable when its ground set has such a cover. Bérczi–Schwarcz–Yamaguchi conjecture. Every $k$-coverable matroid ${\cal M}=(E,\mathcal{I})$ can be reduced to a $2k$-coverable partition matroid on the same ground set $E$. The paper states that this conjecture is refuted by its results, so the proposed universal reduction does not hold.