Suppose I have a matroid $M = (E, \mathcal{I})$. It is a known fact that given any two bases $X_0$ and $X_n$, we can transform $X_0$ into $X_n$ by repeatedly applying the basis exchange axiom. So there is a sequence of bases $X_0, \ldots, X_n$ such that $X_{i+1}$ is obtained by applying basis exchange to $X_i$ and $X_n$.
I am considering an extension of this. Suppose I have two bases $A_0$ and $B_0$ which partition the ground set. That is, $A_0$ and $B_0$ are disjoint bases whose union is the ground set $E$.
Now we are given two other bases $A_n$ and $B_n$. The goal is to transform the partition $A_0, B_0$ into $A_n, B_n$ by repeatedly applying symmetric basis exchange between $A_i$ and $B_i$ to obtain $A_{i+1}$ and $B_{i+1}$. In other words, we are looking for a sequence of partitions $A_0 \cup B_0, \ldots, A_n \cup B_n$ such that $A_i$ and $B_i$ are disjoint bases for each $i$ and $A_i \cup B_i = E$.
Is this always possible? Do we need to make some assumption about the matroid, such as base orderability?
Thank you
This question is equivalent to an open conjecture concerning the connectivity of the basis pair graph. You can find the relevant definitions in the article "Basis Pair Graphs of Transversal Matroids are Connected" by Martin Farber. It has been proven for graphic (Farber-Richter-Shank, 1985), transversal (Farber, 1989), and sparse paving matroids (Bonin, 2013).