The higher-stack equivalence conjecture for Morita linear and abelian categories

For a commutative ring $A$, let $A\textrm{-}\mathcal{CAT}^{Mor}$ denote the Morita localization of linear categories and let $A\textrm{-}\mathcal{AB}^{proj}$ be the subcategory of abelian categories whose morphisms preserve finitely presented projective objects. These assemble into the prefields $\mathcal{CAT}^{Mor}$ and $\mathcal{AB}^{proj}$, with associated higher categorical fields $\overline{\mathcal{CAT}}^{Mor}$ and $\overline{\mathcal{AB}}^{proj}$. Higher-stack equivalence conjecture. The morphism

$$\overline{\mathcal{CAT}}^{Mor}\longrightarrow\overline{\mathcal{AB}}^{proj}$$

should be an equivalence of fields in higher categories. The preceding theorem establishes the analogous equivalence for the full abelian-category prefield at the level of simplicial fields, motivating this conjectural extension to associated higher categorical fields.