Elliptic Fourier-transform compatibility conjecture

Let $G$ be a simple $F$-split group. Let ${\mathcal R}^p_{{\mathrm{un}},{\mathrm{ell}}}(G)$ be the elliptic unipotent representation space, let ${\mathcal C}(G)_{{\mathrm{cpt}},{\mathrm{un}}}$ be the compact unipotent character space, let ${\mathrm{FT}}^\vee_{\mathrm{ell}}$ be the elliptic nonabelian Fourier transform, and let $\operatorname{res}_{{\mathrm{cpt},{\mathrm{un}}}}$ and ${\mathrm{FT}}_{{\mathrm{cpt},{\mathrm{un}}}}$ be the restriction map and compact Fourier transform. Elliptic Fourier-transform compatibility conjecture. The diagram relating these maps commutes up to roots of unity; more precisely, for every unipotent $u\in G^\vee$, elliptic pair $(s,h)\in{\mathcal Y}(\Gamma_u)_{{\mathrm{ell}}}$, and maximal compact open subgroup $K_{\mathcal O}$ of $G$, there is a root of unity $\zeta=\zeta(u,s,h,\mathcal O)$ such that

$$\operatorname{res}_{\mathcal O}(\Pi(u,h,s))=\zeta\cdot({\mathrm{FT}}_{{\mathrm{cpt},{\mathrm{un}}}}\circ\operatorname{res}_{\mathcal O})(\Pi(u,s,h)).$$

This conjecture asserts compatibility between the elliptic nonabelian Fourier transform and the compact Fourier transform; the supplied text gives evidence elsewhere but no resolution status.