Injectivity after restriction to an Iwahori subgroup in semisimple rank zero

Let $G$ be a reductive group with semisimple $F$-rank zero, and let $I$ be an Iwahori subgroup of $G$. Let $\mathfrak{Rep}^{\textnormal{ladm}}(G)$ denote the category of locally admissible representations.

Iwahori-restriction conjecture. If $\pi\in\mathfrak{Rep}^{\textnormal{ladm}}(G)$ is injective, then

$$\pi|_I\in\mathfrak{Rep}^\infty(I)$$

is injective.

This conjecture is proposed in the source as part of the discussion of the semisimple-rank-zero case and is intended to support computations of derived ordinary parts; the supplied text gives no resolution status.