Iwahori component of the Gelfand--Graev representation for reductive groups

Let $G$ be a connected reductive group over a $p$-adic field $F$, $U$ the unipotent radical of a minimal parabolic subgroup, $\psi$ a depth-zero non-degenerate character of $U(F)$, and $I$ an Iwahori subgroup of $G(F)$. We show that, as a module over the Iwahori-Hecke algebra $H$, the space of $I$-fixed vectors in the Gelfand-Graev representation ${\mathrm{i}\mathrm{n}\mathrm{d}}_U^G\psi$ is isomorphic to $H{\otimes }_{H_{W_0}}\mathrm{s}\mathrm{g}\mathrm{n}$. Here $\mathrm{s}\mathrm{g}\mathrm{n}$ is the sign representation of the finite Hecke subalgebra $H_{W_0}$ attached to the relative Weyl group. This extends the theorem of Chan-Savin from split groups to all connected reductive groups.