Alternating path covering conjecture for matroid circuits

Let $M$ be a matroid with ground set $E$ and circuit family $\mathcal C$, and let $A$ and $B$ be bases of $M$. Write $\mathcal C[A\cup B]$ for the circuits contained in $A\cup B$. A graph on $A\triangle B$ covers $\mathcal C[A\cup B]$ if every such circuit contains both endpoints of one of its edges.

Alternating path covering conjecture. There exists a path that alternates between $A\setminus B$ and $B\setminus A$ and covers $\mathcal C[A\cup B]$.

This is presented as a probably overly optimistic conjecture. The paper proves the corresponding covering results for graphic, paving and spike matroids, but the assertion for arbitrary matroids remains open.