Jordan-Hölder property for shifted quantum affine algebras

We prove that finite length representations of shifted quantum affine algebras in category ${\mathcal{O}}^{\mathrm{s}\mathrm{h}}$ are stable by fusion product. This implies that in the topological Grothendieck ring $K_0({\mathcal{O}}^{\mathrm{s}\mathrm{h}})$ the Grothendieck group of finite length representations forms a non-topological subring. We also conjecture this subring is isomorphic to the cluster algebra discovered in arXiv:2401.04616. In the course of our proofs, we establish that any simple representation in category ${\mathcal{O}}^{\mathrm{s}\mathrm{h}}$ descends to a truncation, for certain truncation parameters as conjectured in arXiv:2010.06996 in terms of Langlands dual $q$-characters.