Ultra-generalized basis exchange conjecture

Prove that for any matroid with bases A and B, subsets Xsubseteq A\setminus B and Ysubseteq B\setminus A, and integer p satisfying |Y|\le p\le |X|, there exist subsets Usubseteq A\setminus B and Vsubseteq B\setminus A such that |U\cap X|=p, Y\subseteq V, |U|=|V|\le r((U\cap X)\cup Y), and both A-U+V and B+U-V are bases.

Background

The paper formulates a conjecture intended to simultaneously generalize the authors’ main multiple-exchange theorem and the equitability exchange theorem of Akrami, Raj, and Végh. The authors establish weaker bounds for representable matroids over fields of characteristic zero, but do not prove the conjectured rank bound for arbitrary matroids.

The conjecture is motivated by a far-reaching generalized Grassmann--Plücker identity proved in the paper. The conclusion states that the conjecture was verified computationally for matroids of rank at most five, while the authors’ proof approach does not appear to extend to the conjectured statement.