The classification conjecture for K-matrices on Kirillov–Reshetikhin modules

Let $V$ and $W$ be Kirillov–Reshetikhin modules in the category $\mathcal{C}$, let $\psi$ be a diagram automorphism, and let $K_V(z)$ and $K_W(z)$ satisfy the generalized spectral reflection equation. A symmetrizable invertible solution is an invertible solution of that equation in tensor products of Kirillov–Reshetikhin modules that is symmetrizable in the sense used for the associated tensor-product representation. A grading-shifted universal K-matrix is the action of the universal K-matrix after the grading shift, associated with a quantum symmetric pair having the generalized parameter constraints of the source.

Kirillov–Reshetikhin K-matrix conjecture. Symmetrizable invertible solutions of the generalized spectral reflection equation in tensor products of Kirillov–Reshetikhin modules, with $\psi$ equal to a diagram automorphism, are rescaled actions of the grading-shifted universal K-matrix associated with a quantum symmetric pair with generalized parameter constraints.

The conjecture proposes a classification of spectral reflection-equation solutions in this representation-theoretic setting. It is motivated by known special cases, while the supplied status evidence states that the general description of the corresponding coideal subalgebra remains an open question.