Fourier–Laumon invariance conjecture for cuspidal perverse sheaves

Let $G$ be a possibly disconnected reductive group, let $\mathcal N_G$ be its nilpotent cone, and let $\mathcal O$ be a nilpotent orbit. If $\mathcal E\in\operatorname{Perv}(\mathcal N_G/G)$ is cuspidal, let $\operatorname{infl}(\mathcal E)$ denote its inflation and let $\Four_{\mathfrak g}$ be the Fourier–Laumon transform. Fourier–Laumon invariance conjecture. Then

$$\operatorname{IC}(\mathcal O,\mathcal E)\simeq \Four_{\mathfrak g}\bigl(\operatorname{IC}(\mathcal O\times\mathfrak z(G^\circ),\operatorname{infl}(\mathcal E))\bigr).$$

This strengthens the preceding theorem by predicting that the cuspidal local system can be chosen unchanged under the Fourier transform; the source provides no resolution status.