The folklore irreducibility conjecture for unitary principal series

Let $G$ be a split reductive group over ${\mathbb Q}_p$, with negative Borel subgroup $B^-({\mathbb Q}_p)$, torus $T({\mathbb Q}_p)$, and set of simple roots $S$. Let $\chi:T({\mathbb Q}_p)\rightarrow {\mathcal O}_E^{\times}\subseteq E^{\times}$ be a unitary continuous character, and let $\big(\operatorname{Ind}_{B^-({\mathbb Q}_p)}^{G({\mathbb Q}_p)}\chi\big)^{{\mathcal C}^0}$ denote the corresponding continuous unitary principal series. Folklore irreducibility conjecture. The representation is topologically irreducible if and only if

$$\chi\circ\alpha^{\vee}\ne 1$$

for every $\alpha\in S$. This is described as a folklore conjecture and as a strengthening of a special case of a conjecture of Schraen; the preceding theorem establishes related finite-length, generic irreducibility, and uniqueness results, but not this full assertion.