Higher unramified cohomology extension conjecture

Let $k$ be the base field, let $\mathcal{X}=(X,D)$ be an object of $\category{mSm}$, and let $q\geq 0$. Write $\mathcal{X}^{\circ}$ for the complement of the boundary, $\underline{\mathrm{M}}\mathrm{H}^{q+1}_{\et}$ for the corresponding unramified étale cohomology sheaf, and $\omega^{\mathrm{exc}}\mathrm{H}^{q+1}_{\mathrm{ur}}(\mathcal{X})$ for the exceptional pullback subgroup of $\mathrm{H}^{q+1}_{\mathrm{ur}}(\mathcal{X}^{\circ})$. The sheaf $\underline{\mathrm{M}}\mathrm{H}^{q+1}_{\mathrm{ur}}$ is the resulting unramified cohomology sheaf on $\category{mSm}$. Higher unramified cohomology extension conjecture. For every $q\geq 0$ and $\mathcal{X}=(X,D)\in\category{mSm}$, one has

$$\underline{\mathrm{M}}\mathrm{H}^{q+1}_{\et}(\mathcal{X})=\omega^{\mathrm{exc}}\mathrm{H}^{q+1}_{\mathrm{ur}}(\mathcal{X})$$

as subgroups of $\mathrm{H}^{q+1}_{\mathrm{ur}}(\mathcal{X}^{\circ})$. Moreover, $\underline{\mathrm{M}}\mathrm{H}^{q+1}_{\mathrm{ur}}$ is $(\mathrm{CI}\cup\mathrm{BI})$-local and defines an object $\mathrm{mH}^{q+1}_{\mathrm{ur}}$ of $\category{mDA}^{\eff}(k)$, with an equivalence

$$\Omega_{S^1_t}(\mathrm{mH}^{q+1}_{\mathrm{ur}})\simeq\mathrm{mH}^{q}_{\mathrm{ur}}.$$

The statement is proposed as an extension of the preceding theorem from degree one to all higher unramified cohomological degrees; its resolution is not supplied in the given text.