Khovanov homology in characteristic two and involutive monopole Floer homology

We study the conjugation involution in Seiberg-Witten theory in the context of the Ozsv'ath-Szab'o and Bloom’s spectral sequence for the branched double cover of a link $L$ in $S^3$. We prove that there exists a spectral sequence of $\mathbb{F}[Q]\mathrm{/}{Q}^2$-modules (where $Q$ has degree $-1$) which converges to ${\widetilde{HMI}}_{\ast }(\mathrm{\Sigma }(L))$, an involutive version of the monopole Floer homology of the branched double cover, and whose $E^2$-page is a version of Bar Natan’s characteristic two Khovanov homology of the mirror of $L$. We conjecture that an analogous result holds in the setting of $\mathrm{P}\mathrm{i}\mathrm{n}(2)$-monopole Floer homology.