Unicity conjecture for archetypes of supercuspidal representations of $\mathbf{SL}_N(F)$

Let $F$ be a non-Archimedean local field, let $N$ be a positive integer, and let $\bar{\pi}$ be a supercuspidal representation of $\mathbf{SL}_N(F)$. Choose a supercuspidal representation $\pi$ of $\mathbf{GL}_N(F)$ whose restriction contains $\bar{\pi}$, and define the ramification degree $e_{\bar{\pi}}$ by the condition that there are $N/e_{\bar{\pi}}$ characters $\chi$ of $F^\times$ satisfying

$$\pi\simeq\pi\otimes(\chi\circ\det).$$

This number is independent of the choice of $\pi$. Unicity conjecture. There are precisely $e_{\bar{\pi}}$ archetypes for $\bar{\pi}$, and these archetypes are $\mathbf{GL}_N(F)$-conjugate. The paper presents this as a conjectural extension of its explicit results and as part of a hoped-for positive answer to the unicity question for types; its general proof is left for future work.