The exactness conjecture for the non-linear Fourier transform

Fix the parameter $\underline{\lambda}$ and let $\Psi_{G,\underline{\lambda}}$ be the corresponding gamma $D$-module on $G$. Define the non-linear Fourier transform functor

$$\mathrm F_{G,\underline{\lambda}}:=(-)*\Psi_{G,\underline{\lambda}}:D(G)_{\mathrm{hol}}\to D(G)_{\mathrm{hol}}.$$

The exactness conjecture. The functor $\mathrm F_{G,\underline{\lambda}}$ is exact.

This concerns exactness of convolution with the gamma $D$-module, viewed as a non-linear Fourier transform on holonomic $D$-modules. The supplied passage gives no resolution status.