The Kirillov–Reshetikhin completeness conjecture

Let $\mathfrak g$ be a Lie algebra and let $\mathcal H=\bigotimes_{j=1}^N V_j$ be a tensor product of Kirillov–Reshetikhin modules. Write

$$m_{\lambda,\mathcal H}=\dim\operatorname{Hom}_{\mathfrak g}(\mathcal H,V(\lambda))$$

and let $M_{\lambda,\mathbf n}$ be the number of Bethe-equation solutions determined by the multiplicities $\mathbf n=(n_{\alpha,i})$, where $n_{\alpha,i}$ counts the factors with highest weight $i\omega_\alpha$. Kirillov–Reshetikhin completeness conjecture. The dimension of the space of $\mathfrak g$-linear homomorphisms satisfies

$$m_{\lambda,\mathcal H}=M_{\lambda,\mathbf n}.$$

This is the representation-theoretic form of completeness: the combinatorial count of Bethe solutions should equal the multiplicity of the corresponding highest-weight representation. The source gives no resolution, so the conjecture remains open.