An Ozsváth--Szabó-type spectral sequence for links in $S^1\times S^2$

We show that there is a spectral sequence with $E^2$-page given by the Khovanov homology of a link in $S^1\times S^2$, as defined by Rozansky in arXiv:1011.1958, which converges to the Hochschild homology of an $A_{\mathrm{\infty }}$-bimodule defined in terms of bordered Floer invariants. We also show that the homology algebras $H_{\ast }{\mathfrak{h}}_n$ of the algebras ${\mathfrak{h}}_n$ over which these bimodules are defined give nontrivial $A_{\mathrm{\infty }}$-deformations of Khovanov’s arc algebras $H_n$ for $n>1$.