The weakened Roberts conjecture for cubic Hurwitz stacks

Let $k$ be a field of characteristic not dividing $6$, let $g$ be the genus, and let $\mathrm{Hur}_{3,g,k}$ be the degree-$3$ Hurwitz stack. Write $\mathbb L$ for the class of the affine line and $\widehat{\widetilde{K_0}}(\operatorname{Stacks}_k)$ for the completed Grothendieck ring of stacks. Equality modulo codimension $r$ means that the difference lies in the filtered part of dimension at most the common dimension minus $r$. The weakened Roberts conjecture.

$$\frac{\{\mathrm{Hur}_{3,g,k}\}}{\mathbb L^{\dim \mathrm{Hur}_{3,g,k}}}\equiv (1+\mathbb L^{-1})(1-\mathbb L^{-3})$$

modulo codimension $\frac{g-1}{3}$ in $\widehat{\widetilde{K_0}}(\operatorname{Stacks}_k)$. This is a motivic analogue of the second-order asymptotic for cubic fields, obtained by weakening the codimension predicted by Roberts' conjecture; the source presents it as the next statement to be established in the Grothendieck-ring setting.