Motivic Gersten's conjecture

Let $R$ be a commutative regular local ring, let $\mathcal{M}_R^p$ denote the exact category of finitely generated $R$-modules supported in codimension at least $p$, and let $M_{\#}$, for $\#\in\{\operatorname{add},\operatorname{loc}\}$, be the corresponding additive or localizing motive in $\operatorname{Ho}({\mathcal{M}\!\!ot}^{\#}_{\operatorname{dg}})$. For integers $1\leq p\leq\dim R$, consider the inclusion functor $\mathcal{M}_R^{p-1}\hookrightarrow\mathcal{M}_R^p$.

Motivic Gersten's conjecture. For any commutative regular local ring $R$ and any integers $1\leq p\leq\dim R$, the inclusion functor $\mathcal{M}_R^{p-1}\hookrightarrow\mathcal{M}_R^p$ induces the zero morphism

$$M_{\#}(\mathcal{M}_R^{p-1})\to M_{\#}(\mathcal{M}_R^p)$$

in $\operatorname{Ho}({\mathcal{M}\!\!ot}^{\#}_{\operatorname{dg}})$, where $\#\in\{\operatorname{add},\operatorname{loc}\}$.

This is a motivic strengthening or analogue of the $K$-theoretic Gersten conjecture. The paper presents it as a question in the motivic setting, and the supplied text gives no resolution status.