6 problems

Energy-and-quantum-number completeness conjecture. Eigenstates in the whole Verma module are uniquely identified by their energy, their eigenvalue with respect to…

Uniqueness conjecture. Commutativity with either $\mathsf{I}^{(3)}$ or $\mathsf{K}^{(3)}$ is sufficient to completely fix all the other operators in the integrable structure.

Affine Bethe-vector eigenvector conjecture. The on-shell Bethe vector $\ket{\{t_i\}}_{\mathrm{BA}}$ should be an eigenvector of every local integral of motion $\mathsf{I}^{(2p-1)}$…

Whole-Verma-module simple-spectrum conjecture. For generic $(k,\ell)$, the tuple

Generic simple-spectrum conjecture. For generic $(k,\ell)$, the action of $\mathsf{I}^{(3)}$ generically lifts the degeneracy of the spaces $\Upsilon_{k,\ell}^{(n,m)}$; equivalentl…

Simple-spectrum conjecture. Adding $\mathsf{I}^{(3)}$ lifts the degeneracies of the joint spectrum of $(\mathsf{L}_0,(\vec{\mathsf{J}}_0)^2,\mathsf{J}_0^3)$ and makes the spectrum…