Fourier-Mukai partners that are birational at every point

Question

Let $X$ and $Y$ be smooth projective irreducible varieties that are Fourier-Mukai partners, i.e., have exact equivalent derived categories $D^b(X) \simeq D^b(Y)$. It is well-known that if a derived equivalence $D^b(X) \simeq D^b(Y)$ maps a skyscraper sheaf at $x \in X$ to a skyscraper sheaf at $y \in Y$, then $X$ and $Y$ are birational around those points since the Fourier-Mukai kernel around those points will be a graph of an open immerison. (Moreover, the corresponding birational map will be a $K$-equivalence.) I am wondering if it is true that if for any point $x \in X$, there exists a derived equivalence that maps the skyscraper at $x$ to a skyscrpaer at some point in $Y$ and vice versa, then $X$ and $Y$ are isomorphic. A little bit strongly, is it true that if smooth projective irreducible varieties $X$ and $Y$ are Fourier-Mukai partners and they are birational around every point, they are isomorphic?