Cowling's conjecture on non-archimedean rank-one groups

Let $F$ be a local field of characteristic $0$, let $\mathbf{G}$ be an almost simple, connected and simply connected $F$-algebraic group, and set $G=\mathbf{G}(F)$. In Theorem~, when $F$ is non-archimedean and $\operatorname{rank}_F(\mathbf{G})=1$, there are additional assumptions that $\mathbf{G}$ be $F$-isomorphic to $\operatorname{SL}_2$ or to the $F$-group $\operatorname{SU}_3$ associated with a quadratic extension of $F$. Cowling's conjecture. Those additional conditions regarding non-archimedean rank-one groups could be dropped. This would extend the stated characterization of closed subsets of the unitary dual to all non-compact groups in the theorem's ambient class; the source gives no resolution.