Orbit-method compatibility conjecture for Deligne–Lusztig representations

Let $\Gamma=({\mathcal G}_r^+)^F$, let $\mathcal U=({\mathbb T}_r^+)^F$, and write $\mathfrak g$ and $\mathfrak t$ for their relevant Lie algebras, with $\delta:\mathfrak g\twoheadrightarrow\mathfrak t$ the canonical projection and $\delta^\ast:\mathfrak t^\ast \ifbool{@display}{\longrightarrow}{\rightarrow} \mathfrak g^\ast$ its dual. Let $R_{\rm log}:\mathfrak t^\ast \ifbool{@display}{\longrightarrow}{\rightarrow} \widehat\Gamma$ be the map obtained from the logarithm identification and the cohomological construction, and let $\rho:\mathfrak g^\ast \ifbool{@display}{\longrightarrow}{\rightarrow} \widehat\Gamma$ be the orbit-method map.

Orbit-method compatibility conjecture. We have

$$\rho\circ\delta^\ast=R_{\rm \log}.$$

This conjecture asks whether the cohomological induction map and the orbit-method parametrization of irreducible representations agree after pulling back along the canonical projection. The supplied context presents this as a proposed relation and gives no resolution.