  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
$A_\infty$-bimodule defined in terms of bordered Floer invariants. We also show
that the homology algebras $H_*\mathfrak{h}_n$ of the algebras $\mathfrak{h}_n$
over which these bimodules are defined give nontrivial $A_\infty$-deformations
of Khovanov's arc algebras $H_n$ for $n>1$.
