Classification of irreducible modules in a Schur–Weyl category

Let ${\cal SW}(V)$ be the Schur–Weyl category, let $V_{\lambda}$ denote its distinguished objects, and let ${\bf g}=sl(V_S)$ act on these objects. A linear map between two objects of ${\cal SW}(V)$ is an $sl(V_S)$-morphism when it commutes with this action. The Schur–Weyl module classification conjecture. All the ${\bf g}$-modules $V_{\lambda}$ are irreducible, every irreducible finite-dimensional ${\bf g}$-module is isomorphic to one of them, and a linear map between two objects of ${\cal SW}(V)$ is a morphism in the category exactly when it is an $sl(V_S)$-morphism. Thus the category ${\cal SW}(V)$ should be treatable as the category of $sl(V_S)$-modules, or equivalently of $U(sl(V_S))$-modules. The statement proposes an identification of the Schur–Weyl category with the finite-dimensional representation category of the twisted Lie algebra; the supplied text does not establish the classification or indicate its resolution.