The augmentation-category derived adjunction conjecture

Let $\mathbb{A}$ be an augmentation category, and let $\mathcal{G}_n^\mathrm{sm}(\textbf{dAff})$ and $\mathbb{A}\text{-}\mathcal{G}_n^\mathrm{sm}(\textbf{dAff})$ denote the homotopy categories of the corresponding derived Artin $(\mathbb{A},n)$-geometric stacks; likewise, let $\mathcal{G}_n^\mathrm{'ét}(\textbf{dAff})$ and $\mathbb{A}\text{-}\mathcal{G}_n^\mathrm{'ét}(\textbf{dAff})$ denote the derived Deligne–Mumford versions. Augmentation-category derived adjunction conjecture. There is an adjunction

$$\mathbb{L}i_!\colon\mathcal{G}_n^\mathrm{sm}(\textbf{dAff})\rightleftarrows\mathbb{A}\text{-}\mathcal{G}_n^\mathrm{sm}(\textbf{dAff}):\mathbb{R}i^*$$

(and, respectively, an adjunction)

$$\mathbb{L}i_!\colon\mathcal{G}_n^\mathrm{'ét}(\textbf{dAff})\rightleftarrows\mathbb{A}\text{-}\mathcal{G}_n^\mathrm{'ét}(\textbf{dAff}):\mathbb{R}i^*.$$

The proposed adjunction would relate ordinary and augmented derived geometric stacks. The source explains that relative categories do not have Quillen equivalences, so proving this requires constructing full Quillen model structures and a Quillen adjunction; it may fail for some augmentation categories.