The discreteness-and-tail conjecture for unitary representations of $\overline{\mathrm{GL}}$

Fix $k$ and let $\tau$ be an irreducible representation of $\mathrm{GL}(k,\mathbb F)$. Consider the triples $(z,k,\tau)$ that correspond to unitary representations of $\overline{\mathrm{GL}}$. Discreteness-and-tail conjecture. The set of $z>0$ with this property has the form

$$\{p^{-l}: l\in\mathbb{Z},\ l\geq m(k,\tau)\}.$$

Thus, for each fixed $(k,\tau)$, the admissible positive parameters form a discrete tail. The supplied status evidence says that the statements in this subsection are proved in Section 5, so this conjecture is solved.