Injectivity after restriction to compact open subgroups

Let ${\mathbf G}$ be a semisimple simply connected algebraic group, let $G$ be its group of $F$-points, and let $K$ be a compact open subgroup of $G$ containing the center of $G$. Let $\mathfrak{Rep}^{\textnormal{ladm}}(G)$ be the category of locally admissible representations.

Compact-subgroup injectivity conjecture. If $\pi\in\mathfrak{Rep}^{\textnormal{ladm}}(G)$ is injective, then $\pi|_K$ is injective in $\mathfrak{Rep}^\infty(K)$. Consequently,

$$\textnormal{H}^i\textnormal{Ord}^G_{P^-}\simeq\textnormal{R}^i\textnormal{Ord}^G_{P^-}$$

for $i>0$.

The source presents this as a conjecture implying the desired comparison of cohomological and derived ordinary parts for more general reductive groups; the supplied text gives no resolution status.