The HP-condition unitarity conjecture for cohomological induction

Let $G$ be a real reductive group with Lie algebra $\mathfrak{g}$ and maximal compact subgroup $K$, and let $\pi$ be an irreducible unitary $(\mathfrak{g},K)$ module with infinitesimal character $\Lambda$ satisfying the HP condition. Let $L(x)$ be the Levi subgroup associated with the relevant KGB element $x$, and let $\pi_{L(x)}$ denote the corresponding representation of $L(x)$. HP-condition unitarity conjecture. The representation $\pi_{L(x)}$ must be unitary. This is proposed as a reverse implication to a direction of Vogan's cohomological-induction theorem in a special setting, motivated by calculations for Dirac series. The precise scope of the special setting and the general status require checking in the source.