Classical work by Casselman shows that for an irreducible admissible representation $\rho$ of $GL_2$ over a non-archimedean field $k$, there is a minimal power $n\geq 0$ of the prime ideal $\mathfrak{p}$ such that $\rho$ has a fixed vector under $(\begin{smallmatrix} * & * \\ \mathfrak{p}^n & 1+\mathfrak{p}^n\end{smallmatrix})\subset GL_2(\mathcal{O}_k)$. Furthermore, the fixed space is 1-dimensional.

Suppose $\rho$ is special or supercuspidal. Then it corresponds by Jacquet-Langlands to an irreducible, admissible $D^\times$-representation $\pi$, where $D$ is the non-split quaternions over $k$.

My question is: Is there a compact open subgroup $K\subset D^\times$ such that $\pi^K$ is one-dimensional?

I know, that $\pi$ is finite dimensional

Classical work by Casselman shows that for an irreducible admissible representation $\rho$ of $GL_2$ over a non-archimedean field $k$, there is a minimal power $n\geq 0$ of the prime ideal $\mathfrak{p}$ such that $\rho$ has a fixed vector under $(\begin{smallmatrix} * & * \\ \mathfrak{p}^n & 1+\mathfrak{p}^n\end{smallmatrix})\subset GL_2(\mathcal{O}_k)$. Furthermore, the fixed space is 1-dimensional.

Suppose $\rho$ is special or supercuspidal. Then it corresponds by Jacquet-Langlands to an irreducible, admissible $D^\times$-representation $\pi$, where $D$ is the non-split quaternions over $k$.

My question is: Is there a compact open subgroup $K\subset D^\times$ such that $\pi^K$ is one-dimensional?

Edit: I think, when $\rho$ is special $\chi\cdot St$, its JL-transfer is just $\chi\circ Nrd$, so this case is easy.