Motivic vanishing-cycle localization conjecture

Let $Z'$ be a smooth scheme with trivial $\mathbb{G}_m$-action, and let $\mathbb{A}_{\operatorname{\mathbb{C}}}^n$ carry a $\mathbb{G}_m$-action with nonnegative weights. Let $Z=Z'\times \mathbb{A}_{\operatorname{\mathbb{C}}}^n$ with the induced $\mathbb{G}_m$-action, and let $f:Z\rightarrow \mathbb{A}_{\operatorname{\mathbb{C}}}^1$ be a $\mathbb{G}_m$-equivariant function, with $\mathbb{G}_m$ acting on the target with weight $s>0$. Motivic vanishing-cycle localization conjecture. There is an equality in $\operatorname{K}^{\hat{\mu}}(\operatorname{St^{aff}}/Z')$

$$q_*\phi_f=\mathbb{L}^{-\dim(Z)/2}([f^{-1}(0)]-[f^{-1}(1)]).$$

Here $f^{-1}(0)$ carries the trivial $\hat{\mu}$-action, while the $\hat{\mu}$-action on $f^{-1}(1)$ is given by the natural $\mu_s$-action; both are considered as varieties over $Z'$ via the projection $q:Z\rightarrow Z'$. Equivalently, there is an identity in $\lim_{n\rightarrow\infty}\operatorname{K}^{\mathbb{G}_m,n}(\operatorname{St^{aff}}/\mathbb{A}_{Z'}^1)$:

$$q_*\phi_f=\mathbb{L}^{-\dim(Z)/2}[Z\xrightarrow{f\times q}\mathbb{A}_{Z'}^1].$$

This conjecture is a localization formula for equivariant motivic vanishing cycles and is recalled in the source as a conjecture from the cited work. Its resolution is not established by the supplied text.