 $$\Pi(G/\mathbb{R})_{\psi}^{\mathrm{ABV}}=\Pi(G/\mathbb{R})_{\psi}^{\mathrm{AJ}}.$$  Let $L^{\Gamma}$ be the extended group defined in (\[eq:extendedgroupL\]), and let $\left({}^{\vee}{L}^{\Gamma},\mathcal{S}_L\right)$ be the $E$-group defined in (\[eq:e-groupL\]). Write $\psi_{{L}}$ for the unique (up to conjugation) Arthur parameter of ${}^{\vee}{L}^{\Gamma}$ satisfying $\psi=\iota_{L,G}\circ\psi_{L}$. Let $\