Unitarity of type E genuine small representations

Determine whether the genuine small representations arising from the lift of the trivial representation for simply laced real groups of type $E$ are unitary.

Background

Earlier work computed the lift of the trivial representation for simply laced, split real semisimple groups. The resulting genuine small representations have infinitesimal character $\rho/2$ and are associated with a specific nilpotent orbit characterized by maximal $\tau$-invariant.

The introduction states that the corresponding representations in types $A$ and $D$ are known to be unitary, whereas the type $E$ case has not been settled. This is a specific unresolved instance of the broader question of whether character lifting preserves unitarity.