The discreteness conjecture for spherical characters

Let $\rho$ be an irreducible unitary representation of $\overline{\mathrm{GL}}$, and let $z(\rho)$ denote its spherical character. The preceding construction gives $0\leq z(\rho)\leq 1$. Discreteness conjecture. The number $z(\rho)$ has the form $p^{-k}$, where $k\in\mathbb{Z}_+$, or $z(\rho)=0$. This asserts that the possible nonzero spherical characters are discrete rather than filling a continuum; the supplied text does not indicate whether the claim has been resolved.