Lamy–Sebag conjecture on extending birational maps to piecewise automorphisms

Let $k$ be an algebraically closed field and let $X$ be a $k$-variety. A piecewise automorphism of $X$ is an automorphism on each member of a finite partition of $X$ into locally closed subvarieties. Let

$$\phi:X\dashrightarrow X$$

be a birational map. Lamy–Sebag conjecture. It is possible to extend the map $\phi$ to a piecewise automorphism of $X$. This conjecture gives a conjectural reformulation of the question of when varieties with equal classes in the Grothendieck ring are piecewise isomorphic. A positive answer to the preceding question on equality in the Grothendieck ring is known to imply this conjecture, so the conjecture is solved in the affirmative according to the supplied status evidence.