Algebraic $K$-theory, cohomotopy $K$-groups, and Koszul duality

Let $A$ be an augmented differential graded algebra over a field $k$ of characteristic zero, and let $A^{!}=\mathbf{R}{\mathrm{H}\mathrm{o}\mathrm{m}}_A(k,k)$ be its Koszul dual algebra. Blumberg and Mandell showed that, under some finiteness conditions of $A$, the derived Koszul duality provides an equivalence between the $K$-theory $K({\mathrm{t}\mathrm{h}\mathrm{i}\mathrm{c}\mathrm{k}}_A(k))$ of the triangulated thick subcategory generated by $k$ and the $K$-theory $K(A^{!})$ of the derived category of perfect $A^{!}$-modules. Combining this equivalence with the Jones-Goodwillie Chern character and the Jones-McCleary isomorphism, we obtain that the $K$-groups $K_n({\mathrm{t}\mathrm{h}\mathrm{i}\mathrm{c}\mathrm{k}}_A(k))$ are a concrete candidate for Loday’s conjectural contravariant $K$-groups.