The generalized Grassmann–Plücker exchange conjecture for matroid bases

Let $A$ and $B$ be bases of a matroid $\mathbf{M}$ with rank function $r$. For any $X\subseteq A\setminus B$ and $Y\subseteq B\setminus A$ and integer $p$ with $|Y| \le p \le |X|$, there should exist $U \subseteq A\setminus B$ and $V \subseteq B\setminus A$ such that

$$|U\cap X|=p,\qquad Y\subseteq V,\qquad |U|=|V|\le r((U\cap X)\cup Y),$$

and $A\setminus U\cup V$ and $B\cup U\setminus V$ are bases. Generalized Grassmann–Plücker exchange conjecture. The stated exchange property holds for every matroid. This would provide a common generalization of the paper's equitability-exchange theorem and its main theorem: it recovers the latter when $p=|X|$ and specializes to the former, with additional upper bounds on $|U|$ and $|V|$, when $p=|X|-1\ge |Y|$.