Strengthened Gabow cyclic ordering conjecture for disjoint bases

Let $M=(S,\mathcal{B})$ be a matroid and let $S=B_1\cup\dots\cup B_k$ be a partition of the ground set into $k$ pairwise disjoint bases. Strengthened Gabow conjecture. The matroid $M$ has a cyclic ordering in which the elements of $B_i$ form an interval for each $i\in[k]$. This strengthens the two-basis interval assertion and is motivated by the fact that such a matroid is uniformly dense; the source presents it as an open strengthening.