Liu–Sebag cut-and-paste conjecture for the Grothendieck ring of varieties

Let $k$ be a field, and let $K_0(\operatorname{Var}_k)$ be the Grothendieck ring of varieties over $k$. Liu–Sebag's cut-and-paste conjecture. If varieties $X$ and $Y$ over $k$ satisfy $[X]=[Y]$ in $K_0(\operatorname{Var}_k)$, then there exist disjoint locally closed subvarieties $X_i\subset X$ and $Y_i\subset Y$ such that

$$X=\bigcup_i X_i,\qquad Y=\bigcup_i Y_i,$$

and $X_i\simeq Y_i$ for every $i$. This asserts that equality in the Grothendieck ring is generated by cutting varieties into pairwise isomorphic locally closed pieces; the paper uses it as a conditional hypothesis for disproving motivic stabilization, and the source gives no resolution.