Motivic stability of commuting tuples in general linear groups

For a positive integer $n$, let $C_n(\operatorname{GL}_r(k))$ denote the variety of commuting $n$-tuples in $\operatorname{GL}_r(k)$. Motivic stability conjecture. The limit

$$\lim_{r\to\infty}\frac{[C_n(\operatorname{GL}_r(k))]}{\mathbb{L}^{\dim C_n(\operatorname{GL}_r(k))}}$$

exists in $\widehat{M_{\mathbb{L}}}$. This is an algebraic analogue of a previously known topological stability result for commuting tuples; the paper states that evidence is provided by proving the conjecture for $n=2$.