Commutation of the nonabelian Fourier transforms on elliptic unipotent representations

Let $G$ be the reductive $p$-adic group and let $\mathsf{R}_{\mathsf{un},\mathsf{ell}}$ denote its elliptic unipotent representation space. Let $\operatorname{FT}^{\vee}_{\mathsf{ell}}$ be the Fourier transform on the elliptic unipotent side, let $\operatorname{res}^{\mathsf{par}}_{\mathsf{un}}$ and $\operatorname{res}^{K_0}_{\mathsf{un}}$ be restriction maps to the unipotent representations of parahoric quotients, and let $\operatorname{proj}^{\mathsf{par}}_{\mathsf{ell}}$ be projection to the elliptic parahoric space. Write $\operatorname{FT}^{\mathsf{par}}$ for the parahoric Fourier transform and $\operatorname{FT}^{K_0}$ for the transform associated with a maximal hyperspecial parahoric $K_0$. Commutation conjecture. The two Fourier transforms commute on the elliptic space:

$$\operatorname{proj}^{\mathsf{par}}_{\mathsf{ell}}\circ \operatorname{res}^{\mathsf{par}}_{\mathsf{un}}\circ \operatorname{FT}^{\vee}_{\mathsf{ell}}=\operatorname{proj}^{\mathsf{par}}_{\mathsf{ell}}\circ \operatorname{FT}^{\mathsf{par}}\circ \operatorname{res}^{\mathsf{par}}.$$

Moreover, if $K_0$ is the maximal hyperspecial parahoric of $G$, then on the elliptic subspace $R_{\mathsf{un}}(G)_{\mathsf{ell}}$ one has

$$\operatorname{res}^{K_0}_{\mathsf{un}}\circ \operatorname{FT}^{\vee}_{\mathsf{ell}}=\operatorname{FT}^{K_0}\circ \operatorname{res}^{K_0}_{\mathsf{un}}.$$

This expected compatibility would relate the nonabelian Fourier transform for elliptic unipotent representations of the $p$-adic group to the Fourier transforms for finite reductive quotients of parahoric subgroups. The supplied text gives no resolution, so the conjecture remains open.