Vakil–Wood conjecture on motivic stabilization of symmetric powers

Let $k$ be a field, let $K_0(\operatorname{Var}_k)$ be the Grothendieck ring of varieties over $k$, and let $\mathbb{L}=[\mathbb{A}^1_k]$. Let $\hat K$ be the completion of $K_0(\operatorname{Var}_k)[\mathbb{L}^{-1}]$ with respect to the dimension filtration. Vakil–Wood's motivic stabilization conjecture. For every connected variety $X$ over $k$, the limit

$$\lim_{n\to\infty}\frac{[\operatorname{Sym}^n X]}{\mathbb{L}^{n\dim(X)}}$$

exists in $\hat K$. This conjecture concerns motivic stabilization of symmetric powers, and the paper gives counterexamples conditional on the cut-and-paste conjecture or on $\mathbb{L}$ not being a zero divisor; its general status is therefore not resolved by the source.