The shifted generic-kernel extension conjecture

Let $V^{(i)}_{k,q^p}$ denote the spectral shift of the Kirillov–Reshetikhin module $V^{(i)}_k$, and let $q^pK^{(i)}_k$ denote the corresponding grading shift of its generic kernel. Let $\mathfrak{o}(M,N)$ be the order of the zero at $z=1$ of the normalized $R$-matrix denominator, and let $\mathop{\mathrm{Ext}}^1_{\widetilde{\Pi}}$ be the first extension group over the generalized preprojective algebra. Shifted extension conjecture. For any $i,j\in I$, $k,l\in\mathbb{Z}_{>0}$, and $p,s\in\mathbb{Z}$,

$$\mathfrak{o}(V^{(i)}_{k,q^p},V^{(j)}_{l,q^s})=\dim_{\Bbbk}\mathop{\mathrm{Ext}}^1_{\widetilde{\Pi}}(q^pK^{(i)}_k,q^sK^{(j)}_l).$$

This is the shifted reformulation of the unrestricted divisor–extension conjecture, using the compatibility of spectral and grading shifts.