Lusztig's denominator conjecture for affine Hecke algebras

Let $W_{\mathrm{aff}}$ be an affine Weyl group, let $\mathbf{H}$ be its affine Hecke algebra over $\mathcal{A}=\mathbb{C}[\mathbf{q}^{1/2},\mathbf{q}^{-1/2}]$, and let $J$ be its asymptotic Hecke algebra with basis elements $t_w$. Let $P_W(\mathbf{q})$ be the Poincaré polynomial of the finite Weyl group $W\subset W_{\mathrm{aff}}$, and let

$$\phi\colon \mathbf{H}\mathrel{\hookrightarrow}J\otimes_{\mathbb{C}}\mathcal{A}$$

be Lusztig's map. Writing

$$^\dagger\circ\phi^{-1}(t_w)=\sum_{x\in W_{\mathrm{aff}}}a_{x,w}T_x,$$

Lusztig's denominator conjecture. For all $x,w\in W_{\mathrm{aff}}$, $a_{x,w}$ is a rational function of $\mathbf{q}$; its denominator is independent of $x$ and, as a function of $w$, is constant on two-sided cells. There exists $N_{W_{\mathrm{aff}}}\in\mathbb{N}$ such that

$$P_W(\mathbf{q})^{N_{W_{\mathrm{aff}}}}a_{x,w}\in\mathcal{A}$$

for all $x,w\in W_{\mathrm{aff}}$. Moreover, there exists $N_{W_{\mathrm{aff}}}\in\mathbb{N}$ such that

$$P_W(\mathbf{q})^{N_{W_{\mathrm{aff}}}}d(\omega)\in\mathcal{A}$$

for every formal degree $d(\omega)$ of a discrete series representation of the specialized algebra $H$. This conjecture formulates uniform denominator control for Lusztig's asymptotic Hecke algebra and formal degrees; the paper proves it in several cases, including the lowest two-sided cell and all cells in types $\tilde{A}_n$, $\tilde{C}_2$, and $\tilde{G}_2$, but the general assertion remains open.