Weakly fair A_q(lambda) conjecture for the Dirac series of U(p,q)

Let $U(p,q)$ be the real unitary group, with complexified Lie algebra $\mathfrak{g}$ and maximal compact subgroup $K$. Let $A_{\mathfrak{q}}(\lambda)$ be a cohomologically induced module, and call it weakly fair when its parameter satisfies the weakly fair range condition. Let $\widehat{G}^{d}$ denote the Dirac series, namely the irreducible unitary $(\mathfrak{g},K)$-modules with nonzero Dirac cohomology. Let

denote the necessary condition for nonzero Dirac cohomology given in Lemma 3.5.1 of the source. **Dirac-series conjecture.** Any Dirac series member of $U(p,q)$ must be a weakly fair $A_{\mathfrak{q}}(\lambda)$ module satisfying

.

The conjecture would constrain all irreducible unitary representations with nonzero Dirac cohomology to the weakly fair cohomologically induced setting. The source presents it after computations through $U(5,5)$, but gives no resolution.