Let $(E,I)$ be a matroid, and let $A,B \in I$ be disjoint independent sets in the matroid. Moreover, let $B_1,\ldots, B_k$ be a partition of $B$. I could not decide if the following is always true. There is a subset $S \subseteq A$ and a partition $A_1,\ldots,A_k$ of $S$ such that for all $i \in \{1,2\ldots,k\}$ it holds that $(A \setminus A_i) \cup B_i \in I$ (is independent in the matroid) and $|A_i| \leq |B_i|$.
This forms a generalized exchange property of matroids, w.r.t. the arbitrary partition $B_1,\ldots, B_k$. Any help proving this property or an idea for a counterexample would be appreciated.