Kim's second adjointness conjecture for principal series functors

Let $G$ be a reductive group with symmetric subgroup $K=G^\theta$, let $P$ be a $\theta$-minimal parabolic with unipotent radicals $N$ and $N^-$, and let $M_K=K\cap P$. Write $\fg\operatorname{-mod}_\chi^H$ for the category of $(\fg,H)$-modules with infinitesimal character $\chi$, let $\Av^{K/M_K}_*$ and $\Av^{K/M_K}_!$ be the principal-series functors, let $\Upsilon$ denote the relevant intertwining functor, and define $\ell_{K/M_K}:=\ell_K\otimes\ell_{M_K}^{-1}$. Kim's second adjointness conjecture. The functors

$$\Av^{K/M_K}_*$$

and

$$(-\otimes\ell_{K/M_K}^{-1})\circ\Av^{K/M_K}_!\circ\Upsilon: \fg\operatorname{-mod}_\chi^{M_K\cdot N^-}\rightrightarrows\fg\operatorname{-mod}_\chi^K$$

are canonically isomorphic. Equivalently, the associated formulations using the canonical diagram of functors, the right adjoint of $\Av^{K/M_K}_*\circ\operatorname{oblv}_{N^-}$, or the Casselman–Jacquet functor $J$ all hold. This conjecture is an adjointness statement for principal series and Casselman–Jacquet functors; the supplied text gives no evidence that it has been resolved.