$\mathrm{D}-\mathrm{m}\mathrm{o}\mathrm{d}({\mathrm{B}\mathrm{u}\mathrm{n}}_G^{\mathrm{I}})$ is Compactly Generated

Drinfeld and Gaitsgory proved that $\mathrm{D}-\mathrm{m}\mathrm{o}\mathrm{d}({\mathrm{B}\mathrm{u}\mathrm{n}}_G)$ is compactly generated. Let ${\mathrm{B}\mathrm{u}\mathrm{n}}_G^{\mathrm{I}}$ be the algebraic stack of principal $G$-bundles on $X$ together with Iwahori level structure at a fixed point $x\in X$. More generally, for a finite collection of points $x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_k\in X$, let ${\mathrm{B}\mathrm{u}\mathrm{n}}_G^{(\mathrm{I};x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_k)}$ be the algebraic stack of principal $G$-bundles on $X$ together with Iwahori level structure at each point $x_j$. We will show that $\mathrm{D}-\mathrm{m}\mathrm{o}\mathrm{d}({\mathrm{B}\mathrm{u}\mathrm{n}}_G^{\mathrm{I}})$ and $\mathrm{D}-\mathrm{m}\mathrm{o}\mathrm{d}({\mathrm{B}\mathrm{u}\mathrm{n}}_G^{(\mathrm{I};x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_k)})$ are compactly generated.