Larsen–Lunts correction conjecture for piecewise isomorphism

Let $k$ be a convenient field, and let $K_0(\mathcal{V}_k)$ denote the Grothendieck ring of varieties over $k$. For varieties $X$ and $Y$, write $[X]$ and $[Y]$ for their classes in this ring, let $\mathbb{A}^1$ be the affine line, and let $\amalg$ denote disjoint union. Two varieties are piecewise isomorphic if they admit finite stratifications by locally closed subvarieties whose corresponding strata are isomorphic.

Larsen–Lunts correction conjecture. If

$$[X]=[Y]$$

in $K_0(\mathcal{V}_k)$, then there exist varieties $X'$ and $Y'$ such that

$$[X']\ne [Y'],\qquad [X'\times\mathbb{A}^1]=[Y'\times\mathbb{A}^1],$$

and $X\amalg(X'\times\mathbb{A}^1)$ is piecewise isomorphic to $Y\amalg(Y'\times\mathbb{A}^1)$.

This conjecture proposes that elements in the kernel of multiplication by the Lefschetz class $\mathbb{L}=[\mathbb{A}^1]$ are the only possible errors preventing equality in the Grothendieck ring from implying piecewise isomorphism.