The Oh–Park denominator conjecture for Kirillov–Reshetikhin modules

Let $I$ index the simple roots, let $d_i$ be the symmetrizing integers, let $r$ and $h^\vee$ be the parameters used in the paper, and let $\widetilde{c}_{ij}(u)$ be the coefficients of the formal Taylor expansion of the inverse deformed Cartan matrix. For Kirillov–Reshetikhin modules $V^{(i)}_k$ and $V^{(j)}_l$, let $d_{V^{(i)}_k,V^{(j)}_l}(z)$ be the monic denominator of the normalized $R$-matrix. Denominator conjecture. For $i,j\in I$ and $k,l\in\mathbb{Z}_{>0}$ with $kd_i\ge ld_j$, one has

$$d_{V^{(i)}_k,V^{(j)}_l}(z)=d_{V^{(j)}_l,V^{(i)}_k}(z)=\prod_{a=0}^{l-1}\prod_{u=0}^{rh^\vee}\left(z-q^{u+kd_i+(2a-l+1)d_j}\right)^{\widetilde{c}_{ij}(u)},$$

unless the condition $(\clubsuit)$ in Lemma~ is satisfied. The formula is known for types $\mathrm{A}_n$, $\mathrm{B}_n$, $\mathrm{C}_n$, $\mathrm{D}_n$, and $\mathrm{G}_2$, and for arbitrary type when $(k,l)=(r/d_i,1)$; the excluded exceptional case reflects gaps in an earlier proof.