The scattered-part cardinality conjecture for type A

Let $N_{A_n}$ denote the cardinality of the scattered part of $\widehat{A}_n^{\mathrm{d}}$, the set of irreducible unitary representations for the complex simple group of type $A_n$ with non-zero Dirac cohomology. The type-A scattered-part conjecture.

$$N_{A_n}=2^{n-1}.$$

This is presented after explicit calculations for ranks $1$ through $6$ and is not accompanied in the supplied text by a proof or a resolution.