Piecewise-isomorphism conjecture for birational complements

Let $k$ be an algebraically closed field of characteristic zero. Let $X$ be a separated scheme of finite type over $k$, and let $f\colon X\dasharrow X$ be a birational map inducing an isomorphism of $k$-schemes $U\cong V$ on dense open subsets $U,V$ of $X$. Recall that two $k$-schemes of finite type are piecewise isomorphic if each admits a finite partition into locally closed subschemes whose corresponding reduced subschemes are isomorphic over $k$.

Piecewise-isomorphism conjecture. The $k$-schemes

$$(X\setminus U)_{\mathrm{red}}\quad\text{and}\quad (X\setminus V)_{\mathrm{red}}$$

are piecewise isomorphic.

This conjecture is a reformulation of the question whether equality in the Grothendieck ring of varieties implies piecewise isomorphism. It holds for separated schemes of finite type of dimension at most $1$ over an algebraically closed field of characteristic zero, but remains open in general, particularly for schemes having infinitely many rational curves.