Finkelberg–Ginzburg's mirabolic monodromy conjecture

Let $H=H^{\mathrm{SL}_n}$ be the maximal torus of $SL_n(\mathbb C)$, let $\vartheta\in\mathfrak h^*/W$, and set $k=c-1$. Let $\nabla^{K}(\vartheta,k)$ be the trigonometric KZ connection with values in the covariant representation $\operatorname{K}_\vartheta$ of the degenerate affine Hecke algebra $\operatorname{H_k}$. Let $H^\vee=\operatorname{Hom}_{\mathbb Z}(P^\vee,\mathbb C^\times)$, set $\Theta=\exp_{H^\vee}(\vartheta)\in H^\vee/W$, and let $\operatorname{K}(\Theta)$ be the covariant representation of the extended affine Hecke algebra $\operatorname{H_{q}^{\mathrm{ext}}}$, where $q=\exp(2\pi \mathrm{i}k)$. Assume that $c=k+1$ is not a rational number of the form $\frac{p}{m}$, where $2\leq m\leq n$, $1\leq p\leq m$, and $\mathrm{gcd}(p,m)=1$. Finkelberg–Ginzburg's mirabolic monodromy conjecture. The monodromy of the trigonometric KZ connection $\nabla^{K}(\vartheta,k)$ with values in the covariant representation $\operatorname{K}_\vartheta$ of $\operatorname{H_k}$ is isomorphic to the covariant representation $\operatorname{K}(\Theta)$ of the extended affine Hecke algebra $\operatorname{H_{q}^{\mathrm{ext}}}$. This identifies the generic monodromy of the trigonometric KZ connection with the expected extended affine Hecke algebra representation, while the excluded rational parameters are precisely the exceptional cases not covered by the conjecture. The conjecture is attributed to Finkelberg and Ginzburg.