Let $M$ be a matroid and $A,B$ be two bases of $M$. The basis exchange property tells us that for every $a\in A\setminus B$, there exists $b\in B\setminus A$ such that $(A\setminus \{a\})\cup \{b\}$ is a basis of $M$.
I wonder if the following statement is true:
There exists $a\in A$ and $b\in B$ such that both $(A\setminus \{a\})\cup \{b\}$ and $(B\setminus \{b\})\cup \{a\}$ are bases of $M$ (i.e., we swap $a$ and $b$).
If not, do we have a counterexample?
Or even stronger: what if we just let $A,B$ to be two independent sets with the same cardinality?