The Khovanov-to-annular Khovanov spectral sequence conjecture for 2-periodic links

Let $\tilde L$ be a 2-periodic link in $S^3$ with quotient link $L$. Write $Kh$ for Khovanov homology and $AKh$ for annular Khovanov homology, both over $\mathbb{F}_2$, and let $\theta$ be an invertible formal variable. Khovanov-to-annular Khovanov spectral sequence conjecture. There is a spectral sequence with

$$E^1 \cong Kh(\tilde L) \otimes_{\mathbb{F}_2} \mathbb{F}_2[\theta,\theta^{-1}] \rightrightarrows E^\infty \cong AKh(L) \otimes_{\mathbb{F}_2} \mathbb{F}_2[\theta,\theta^{-1}].$$

This would imply the cascade

$$\operatorname{rk}_{\mathbb{F}_2} AKh(\tilde L) \geq \operatorname{rk}_{\mathbb{F}_2} Kh(\tilde L) \geq \operatorname{rk}_{\mathbb{F}_2} AKh(L) \geq \operatorname{rk}_{\mathbb{F}_2} Kh(L),$$

where the first and third inequalities are given by the $k$-grading filtration on $CKh(\mathcal{D}(\tilde L))$ and $CKh(\mathcal{D}(L))$. The conjecture is motivated by the Khovanov–Tate bicomplex and is supported by analogous results for certain families of links; its general validity remains open.