Frenkel–Mukhin conjecture for the q-character algorithm

Let $U_q(\hat{\mathfrak{g}})$ be a quantum affine algebra, let $m_+$ be a dominant monomial in $\mathbb{Z}[Y_{i,a}^{\pm1}]_{i\in I;\,a\in\mathbb{C}^{\times}}$, and let $V(m_+)$ be the irreducible representation with highest weight monomial $m_+$. The Frenkel–Mukhin algorithm produces a polynomial $\chi(m_+)$.

Frenkel–Mukhin conjecture. For any dominant monomial $m_+$, the algorithm never fails and stops after finitely many steps. Moreover,

$$\chi(m_+)=\chi_q(V(m_+)).$$

This conjecture asserts that the algorithm gives an effective formula for the $q$-character of every irreducible representation. The source provides no resolution status.