The refined Adams conjecture for ABV-packets

Let $G=G(W_n)$ and $H^\pm=H(V_m^\pm)$ be members of the relevant classical dual-pair towers, and let $\phi\in\Phi(G)$. Define $\phi_\alpha$ by

$$\phi_\alpha=(\chi_W\chi_V^{-1}\otimes{}^c\phi^\vee)\oplus\left(\bigoplus_{i=0}^{\alpha-1}\chi_W|\cdot|^{\frac{\alpha-1}{2}-i}\otimes S_1\right).$$

The refined Adams conjecture for ABV-packets. If $\pi\in\Pi_\phi^{\mathrm{ABV}}$, then: (1) for $\alpha\gg0$, $\theta_{-\alpha}^\pm(\pi)\in\Pi_{\phi_\alpha}^{\mathrm{ABV}}$; (2) if $\theta_{-\alpha}^+(\pi)\neq0$, then $\theta_{-\alpha}^\pm(\pi)\in\Pi_{\phi_\alpha}^{\mathrm{ABV}}$; (3) if $\theta_{-\alpha}^\pm(\pi)\in\Pi_{\phi_\alpha}^{\mathrm{ABV}}$ for some $\alpha$, then $\theta_{-(\alpha+2)}^\pm(\pi)\in\Pi_{\phi_{\alpha+2}}^{\mathrm{ABV}}$; (4) if $\theta_{-\alpha}^\pm(\pi)\in\Pi_{\phi_\alpha}^{\mathrm{ABV}}$, then $\theta_{-\alpha}^\pm(\pi)\in\Pi_{(\phi_\pi)_\alpha}^{\mathrm{ABV}}$; and (5), if $\pi\in\Pi_\phi^{\mathrm{ABV}}\cap\Pi_{\phi'}^{\mathrm{ABV}}$ with $\phi\geq_C\phi'$, then $\theta_{-\alpha}^\pm(\pi)\in\Pi_{(\phi')_\alpha}^{\mathrm{ABV}}$ implies $\theta_{-\alpha}^\pm(\pi)\in\Pi_{\phi_\alpha}^{\mathrm{ABV}}$.