Intertwining-operator conjecture for $\rho_c(\tau)$

Let $\tau$ be a genuine unitary irreducible generic representation of $\operatorname{GL}_k^{(m)}$. For an integer $l\geq1$ and $\zeta\in\mathbb{C}^{l}$, let

$$M(\zeta,w_{(k^{l})}):\operatorname{Ind}_{\widetilde{P}_{(k^{l})}}^{\operatorname{GL}_{lk}^{(m)}}\left(\bigotimes_{i=1}^{l}|\det|^{\zeta_i}\tau\right)\longrightarrow\operatorname{Ind}_{\widetilde{P}_{(k^{l})}}^{\operatorname{GL}_{lk}^{(m)}}\left(\bigotimes_{i=1}^{l}|\det|^{\zeta_{l-i+1}}\tau\right),$$

be the standard intertwining operator, and set

$$\zeta^{(l)}=\left(\frac{l-1}{2r},\frac{l-3}{2r},\ldots,\frac{1-l}{2r}\right)\in\mathbb{C}^{l}.$$

Intertwining-operator conjecture. For all $1<l\leq r$, $M(\zeta,w_{(k^{l})})$ is well defined at $\zeta=\zeta^{(l)}$ and has irreducible image. For all $c\geq1$, $M(\zeta,w_{(k^{rc})})$ is well defined at $\zeta=\zeta^{(rc)}$, and its image is irreducible and isomorphic to $\rho_c(\tau)$. In particular, it is $(rk,c)$.

The claim is known in the cases $r=1$ and $k=1$ cited by the source; the general assertion concerns the irreducibility and identification of these specialized intertwining-operator images.