Serial symmetric exchanges

Definitions and mathematical statement

Let $M$ be a finite matroid and $A,B$ two bases. Recall that matroid independence is hereditary and satisfies exchange, and that a basis is a maximal independent set. Fix $X\subseteq A$ with $|X|=m$. A serial symmetric exchange consists of a set $Y\subseteq B$, $|Y|=m$, and orderings $x_1,\ldots,x_m$ of $X$ and $y_1,\ldots,y_m$ of $Y$ such that, for every $0\le i\le m$, both \[ (A\setminus\{x_1,\ldots,x_i\})\cup\{y_1,\ldots,y_i\}, \qquad (B\setminus\{y_1,\ldots,y_i\})\cup\{x_1,\ldots,x_i\} \] are bases.

Must such a choice exist for every matroid, every pair of bases, and every subset $X$ of the first basis? The two lists may be ordered in whichever way is most favorable; the question does not prescribe their order in advance.

The same paired sequence must keep both evolving sets as bases at every intermediate stage. A simultaneous exchange of the full sets at the final stage alone is weaker, as is maintaining only one of the two bases during the process. The bases $A,B$ may overlap; their elements are treated as actual ground elements, not automatically as disjoint copies. In particular a common element may occur in both lists when consistent with the displayed basis conditions. The empty-subset case requires no exchanges.

Short English statement

Can any chosen subset of one matroid basis be exchanged with an equally sized subset of another, one pair at a time, keeping both sets bases throughout?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3000077 (AMR-029-0077), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Egres Open - List of open problems, Open-problem page: Serial symmetric exchanges. Original source indexed by AMR-029-0077. https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems.
• S3 EGRES Open, Serial symmetric exchanges. Individual source page and explanatory remarks. https://lemon.cs.elte.hu/egres/open/Serial_symmetric_exchanges.