The maximal and minimal local Arthur parameter conjecture

Let $G$ be a connected reductive group defined over a non-Archimedean local field, and assume that a theory of local Arthur packets is available. For an irreducible representation $pi e Pi_A(G)$, let $Psi(pi)$ be the set of local Arthur parameters associated with $pi$, and let $lambda_{phi_psi}$ denote the infinitesimal parameter associated with the local $L$-parameter $phi_psi$.

Maximal and minimal parameter conjecture. For any $psi_1,psi_2 e Psi(pi)$, one has $lambda_{phi_{psi_1}}=lambda_{phi_{psi_2}}$. Moreover, there are unique $psi^{max}(pi),psi^{min}(pi) e Psi(pi)$ such that

$$\psi^{\max}(pi) \geq_C \psi \geq_C \psi^{\min}(pi)$$

for every $psi e Psi(pi)$.

The conjecture extends the maximal and minimal parameter characterization beyond classical groups, conditional on the existence of local Arthur packets. Its status for general connected reductive groups is open.