Strong nonvanishing conjecture for normalized intertwining operators

Let $\tau_a$ and $\tau_b$ be the representations, and let $\rho_c(\tau_a)$ and $\rho_d(\tau_b)$ be the associated representations, with $M$, $\bar{s}$, and $b1$ as in the Main Theorem. Define

$$M^*(s,\rho_c(\tau_a)\otimes \rho_d(\tau_b)):=\frac{1}{\alpha(s,\rho_c(\tau_a),\rho_d(\tau_b))}M(\sigma,\rho_c(\tau_a)\otimes \rho_d(\tau_b),\bar{s}).$$

Strong nonvanishing conjecture. The operator $M^*(s,\rho_c(\tau_a)\otimes \rho_d(\tau_b))$ is nonzero for every $s\in\mathbb{C}$.

This conjecture proposes a strong form of the paper's Main Theorem and would imply a complete answer to the reducibility problem. The reducibility problem was already settled by Tadić and Lapid–Mínguez through a combinatorial analysis of Jacquet modules, so the conjecture's status should be checked against those results.