The odd-prime localization conjecture for symplectic annular Khovanov homology

Let $L$ be a $p$-periodic link for an odd prime $p$, and let $\overline{L}$ be its quotient link, treated as an annular link with the axis of symmetry as the annular axis. Write $H^*(B\mathbb Z_p;\mathbb F_p)=\mathbb F_p[\alpha,\theta]/(\alpha^2)$, where $\deg(\alpha)=1$ and $\deg(\theta)=2$, and let $\theta^{-1}H^*(B\mathbb Z_p;\mathbb F_p)$ denote the corresponding localization. The odd-prime localization conjecture. There is a spectral sequence with $E_1$ page isomorphic to

$$\operatorname{Kh}_{symp}(L;\mathbb F_p)\otimes \theta^{-1}H^*(B\mathbb Z_p;\mathbb F_p)$$

and $E_\infty$ page isomorphic to

$$\operatorname{AKh}_{symp}(\overline{L};\mathbb F_p)\otimes \theta^{-1}H^*(B\mathbb Z_p;\mathbb F_p).$$

This would extend the established 2-periodic localization result to odd prime-periodic links. It is motivated by expected fixed-point localization for Lagrangian Floer cohomology with a $\mathbb Z/p\mathbb Z$-action, but such a theorem is not presently available in the paper's setting.