Affine Macdonald denominator conjecture

Let $\mathsf{k}\geq 0$. For the affine Macdonald trace function associated with the zero weight and level zero, $\chi_{0,0,\mathsf{k}}(q,\lambda,\omega)$, let $\rho$ denote the appropriate Weyl vector, let $d$ be the grading element, and let $\operatorname{mult}(\alpha)$ denote the multiplicity of a positive affine root $\alpha$. The notation $(q^a;q^b)$ denotes the $q$-Pochhammer symbol.

Affine denominator conjecture. The affine Macdonald denominator is given by

$$\chi_{0,0,\mathsf{k}}(q,\lambda,\omega)=q^{2(\mathsf{k}-1)(\rho,\lambda)}\frac{\prod_{i=1}^{\mathsf{k}-1}(q^{-2\omega+2i};q^{-2\omega})}{\prod_{i=1}^{\mathsf{k}-1}(q^{-2\omega+2ni};q^{-2\omega})}\prod_{i=1}^{\mathsf{k}-1}\prod_{\alpha>0}\left(1-q^{-2(\alpha,\lambda+\omega d)+2i}\right)^{\operatorname{mult}(\alpha)}.$$

This is presented as a refinement of Etingof–Kirillov Jr.'s affine Macdonald denominator conjecture, but the supplied material gives no resolution status; the formula is intended to provide an affine analogue of the Macdonald denominator.