Gross–Reeder root number conjecture for adjoint gamma-factors

Let $T\subset G$ be a maximal torus split over $F$, with root datum $(X(T),\Phi(T),X^{\vee}(T),\Phi^{\vee}(T))$, and let $G^\sphat$ be the dual group. Define

$$2\rho=\sum_{0<\alpha\in\Phi^{\vee}(T)}\alpha,$$

and let $\epsilon=2\rho(-1)\in T$ be the resulting central element of $G$. For a discrete parameter $\varphi$, write $w(\varphi,\operatorname{Ad})$ for the root number of the adjoint representation, and let $\pi$ be the representation associated with $\varphi$. Gross–Reeder's root number conjecture. One should have

$$\frac{w(\varphi,\operatorname{Ad})}{w(\varphi_0,\operatorname{Ad})}=\pi(\epsilon).$$

This conjecture predicts the variation of adjoint root numbers across the local Langlands correspondence in terms of the central element determined by the root datum; the source gives no resolution status.