Integral unitary representations conjecture

Let $G$ be a reductive group with Langlands parameter $\varphi$ satisfying $I_F\subseteq\ker\varphi$ and

$$\varphi(\mathsf{Fr})=q^\lambda,$$

where $\lambda$ is dominant integral in the fundamental positive chamber. Let $P=MN$ be the parabolic subgroup defined by $\lambda$, and let $M=G(\lambda)$ act on

$$V=\mathfrak g_1=\mathfrak n/[\mathfrak n,\mathfrak n].$$

Assume that $V$ has finitely many $M$-orbits and that $\pi(\mathcal E)\in\Pi(\varphi)$ is parametrised by a simple $M$-equivariant perverse sheaf $\mathcal E$. Integral unitary representations conjecture. The representation $\pi(\mathcal E)$ is unitary if and only if there exists an Arthur pair $({\mathcal O},{\mathcal O}')$ of $M$-orbits in $V$ such that $\overline{T^*_{\mathcal O}V}$ occurs in the characteristic cycle $CC(\mathcal E)$. This conjecture relates unitarity of integral Langlands representations to the geometry of characteristic cycles and Arthur pairs. Its spherical special case was verified using the classification of the spherical unitary dual, while the general equivalence is resolved in the cited work.