Let $G$ be a split reductive group over $\mathbf{Q}_p$ and assume $G$ has connected center. Let $T$ be a maximal split subtorus of $G$ and $R$ be the roots of $(G,T)$.

Let $\chi : T(\mathbf{Q}_p) \to \mathbf{Z}_p^\times$ be a continuous character and assume $\chi \circ \alpha^\vee \neq 1$ for all $\alpha \in R$.

Question : Do we have $w(\chi)=\chi$ if and only if $w=1$ ?

This is true for $\mathrm{GL}_n$ or $\mathrm{GSp}_{2n}$ and also for unramified characters, but is it in general ?

Remark : If the center of $G$ is not connected there are counterexamples, e.g. $G=\mathrm{SL}_2$ and $\chi : \mathrm{diag}(x,x^{-1}) \mapsto (-1)^{\mathrm{ord}_p(x)}$.