The closure-monotonicity conjecture for Adams lifts

Let $G=G(W_n)$ and let $\psi,\psi'\in\Psi^+(G_n)$ be local Arthur parameters with $\psi\geq_C\psi'$. Suppose that $\pi\in\Pi_\psi\cap\Pi_{\psi'}$. The closure-monotonicity conjecture. If $\theta_{-\alpha}^\pm(\pi)\in\Pi_{\psi'_\alpha}$, then $\theta_{-\alpha}^\pm(\pi)\in\Pi_{\psi_\alpha}$.