Ultra-generalized exchange for valuated matroids

Establish that the ultra-generalized exchange property holds for arbitrary valuated matroids: for subsets X and Y and an integer p with |Y|\le p\le |X|, find U and V satisfying |U\cap X|=p, Y\subseteq V, |U|=|V|\le p+|Y|, and \omega(A)+\omega(B)\le\omega(A-U+V)+\omega(B+U-V).

Background

The paper proves the stated exchange inequality for representable valuated matroids arising from matrices over valuated fields of characteristic zero with valuation trivial on the rationals. It then conjectures that the same property should hold for all valuated matroids, a broader class not necessarily arising from such matrix representations.

This unresolved extension would generalize the paper’s representable-valuated result and could support corresponding extensions of equitability results to arbitrary valuated matroids.