Galkin–Kontsevich weak zero-divisor conjecture for the Grothendieck ring

Let $\bfk$ be an algebraically closed field of characteristic $0$, let $d$ be the dimension of a cubic hypersurface, and let $K_0(\operatorname{Var}/\bfk)$ be the Grothendieck ring of algebraic varieties over $\bfk$. Write $\bL=[\mathbb A^1]$ for the class of the affine line. A class $\alpha$ is represented by a linear combination of varieties of dimension at most $2(d-2)$ if it is represented by an element of $K_0(\operatorname{Var}/\bfk)$ with such a representation.

Galkin–Kontsevich weak conjecture. If $\alpha$ is represented by a linear combination of varieties of dimension less than or equal to $2(d-2)$ and

$$\alpha\cdot \bL^2=0,$$

then

$$\alpha\in\langle \bL\rangle\subset K_0(\operatorname{Var}/\bfk),$$

where $\langle\bL\rangle$ is the principal ideal generated by $\bL$.

This is a slightly weaker version of the conjecture of Galkin and Kontsevich mentioned in the source. It was proposed because it would imply the irrationality of generic $d$-dimensional cubic hypersurfaces; the source does not provide a resolution of this weaker statement.