Ayoub's contraction conjecture for n-motives

Let $\boldsymbol{k}$ be a perfect field, let $R$ be a $\mathbb{Q}$-algebra, and let $\mathrm{DM}^{\mathrm{eff}}_{\acute et}(\boldsymbol{k},R)$ be the effective étale motivic category. Let $\mathrm{DM}^{\mathrm{eff}}_{\acute et,\leq n}(\boldsymbol{k},R)$ be its localizing subcategory generated by motives of smooth schemes of dimension at most $n$. Let $\underline{\operatorname{Hom}}^{\mathrm{eff}}$ denote the internal hom and let $R(1)$ be the Tate motive. Ayoub's contraction conjecture. The functor

$$\underline{\operatorname{Hom}}^{\mathrm{eff}}(R(1),-):\mathrm{DM}^{\mathrm{eff}}_{\acute et}(\boldsymbol{k},R)\longrightarrow\mathrm{DM}^{\mathrm{eff}}_{\acute et}(\boldsymbol{k},R)$$

takes $\mathrm{DM}^{\mathrm{eff}}_{\acute et,\leq n}(\boldsymbol{k},R)$ to $\mathrm{DM}^{\mathrm{eff}}_{\acute et,\leq n-1}(\boldsymbol{k},R)$. This is one of Ayoub's conjectures governing the behavior of $n$-motives under contraction by the Tate motive; it is presented as conjectural in the supplied text.