# Serial symmetric exchanges

Let M be a matroid, and let A and B be two bases of M. A subset X of A and a subset Y of B, both of size k, form a serial symmetric exchange with respect to A and B if there is an ordering $x_1,\dots,x_k$ of X and an ordering $y_1,\dots,y_k$ of Y such that both $A \setminus \{x_1,\dots,x_i\} \cup \{y_1,\dots,y_i\}$ and $B \setminus \{y_1,\dots,y_i\} \cup \{x_1,\dots,x_i\}$ are bases for every $i \in \{1,\dots,k\}$.
Is it true that for any matroid M, any two bases A and B, and any $X \subseteq A$, there exists $Y \subseteq B$ such that X and Y form a serial symmetric exchange with respect to A and B?
This conjecture was proposed by Kotlar and Ziv [1]. If true, it would imply the conjecture on Cyclic orderings of matroids for k=2, and the conjecture on Equitability of matroids. Kotlar and Ziv proved that the conjecture is true if $|X|=2$.