Extension by zero and restriction for magic categories

Let $\mathcal{Y}$ be a smooth stack, let $\mathcal{S}\subset\mathcal{Y}$ be the relevant closed stratum, and let $j\colon\mathcal{Y}^{\circ}=\mathcal{Y}\setminus\mathcal{S}\hookrightarrow\mathcal{Y}$ be the open immersion. Let $\operatorname{M}(\mathcal{Y})_\delta$ and $\operatorname{M}(\mathcal{Y}^{\circ})_\delta$ be the corresponding magic categories, and let $\Upsilon\colon\operatorname{M}(\mathcal{Z})_{\delta'}\to\operatorname{M}(\mathcal{Y})_\delta$ be the functor associated with the stratum. Magic-category adjoint conjecture. The restriction functor admits both fully faithful adjoints, with semiorthogonal decompositions

$$\operatorname{M}(\mathcal{Y})_\delta=\left\langle j_*\operatorname{M}(\mathcal{Y}^{\circ})_\delta,\Upsilon(\operatorname{M}(\mathcal{Z})_{\delta'})\right\rangle$$

and

$$\operatorname{M}(\mathcal{Y})_\delta=\left\langle \Upsilon(\operatorname{M}(\mathcal{Z})_{\delta'}),j_!\operatorname{M}(\mathcal{Y}^{\circ})_\delta\right\rangle.$$

The paper proves this conjecture for quotient stacks, while the general formulation is proposed before that result.