Affine Macdonald evaluation conjecture

Let $J_{\mu,k,\mathsf{k}}(q,\lambda,\omega)=\chi_{\mu,k,\mathsf{k}}(q,\lambda,\omega)/\chi_{0,0,\mathsf{k}}(q,\lambda,\omega)$ be the affine Macdonald polynomial for $U_q(\widehat{\mathfrak{sl}}_n)$, with $\rho$ the finite Weyl vector, $\widetilde{\rho}$ the affine Weyl vector, and $\operatorname{mult}(\alpha)$ the multiplicity of a positive affine root. Affine Macdonald evaluation conjecture. For $|q|>1$, one has

$$J_{\mu,k,\mathsf{k}}(q,\mathsf{k}\rho,\mathsf{k}n)=q^{2(\mu,\mathsf{k}\rho)}\frac{\prod_{i=1}^{\mathsf{k}-1}(q^{-2i};q^{-2(k+\mathsf{k}n)})}{\prod_{i=1}^{\mathsf{k}-1}(q^{-2ni};q^{-2(k+\mathsf{k}n)})}\frac{\prod_{i=1}^{\mathsf{k}-1}(q^{-2ni};q^{-2\mathsf{k}n})}{\prod_{i=1}^{\mathsf{k}-1}(q^{-2i};q^{-2\mathsf{k}n})}\prod_{\alpha>0}\prod_{i=0}^{\mathsf{k}-1}\frac{(1-q^{-2(\alpha,\mu+k\Lambda_0+\mathsf{k}\widetilde{\rho})-2i})^{\operatorname{mult}(\alpha)}}{(1-q^{-2(\alpha,\mathsf{k}\widetilde{\rho})-2i})^{\operatorname{mult}(\alpha)}}.$$

The formula modifies the earlier evaluation conjecture so that it converges in the region required by the trace. Its general validity remains open.