The canonical-solution conjecture for KR $Q$-systems

Let $\mathfrak{g}_0$ be the finite-dimensional Lie algebra associated with $U_q(X_N^{(r)})$, let $H$ be the index set of pairs $(a,m)$ for the KR $Q$-system, and let $\mathcal{Q}_m^{(a)}(y)$ be the normalized $\mathfrak{g}_0$-character of the KR module $W_m^{(a)}(\zeta)$. Let $G$ be the matrix specified by the KR-type $Q$-system. Canonical-solution conjecture. The family $(\mathcal{Q}_m^{(a)}(y))_{(a,m)\in H}$ is characterized as the canonical solution of the KR-type $Q$-system with matrix $G$. This is the paper’s main reformulation of the Kirillov–Reshetikhin conjecture; the supplied text gives no resolution, so the assertion remains open.