The local Shimura conjecture for standard intertwining operators

Let $F$ be the relevant number field, let $\nu$ be a place of $F$, and let $\tau_\nu$ be the local representation used to form the induced representations. For $l\geq 2$, write $\boldsymbol{\zeta}=(\zeta_1,\ldots,\zeta_l)\in\mathbb{C}^l$, and let $w_l$ be the Weyl element interchanging the $l$ blocks of size $k$. Let $M(w_l,\boldsymbol{\zeta})$ be the standard intertwining operator from the induction with inducing data $|\det|^{\zeta_1}\tau_\nu\otimes\cdots\otimes|\det|^{\zeta_l}\tau_\nu$ to the induction with the reversed inducing data. Local Shimura conjecture. For each place $\nu$, $M(w_2,\boldsymbol{\zeta})$ is holomorphic for $\operatorname{Re}(\zeta_1-\zeta_2)\geq 1/r$, and if $1<l\leq r$, the image of $M(w_l,\boldsymbol{\zeta})$ is irreducible at

$$\boldsymbol{\zeta}=((l-1)/(2r),(l-3)/(2r),\ldots,(1-l)/(2r)).$$

This is a local analytic and irreducibility assertion for the intertwining operators underlying the doubling construction; the supplied material gives no resolution status beyond calling it a local conjecture.