  We define a family of Khovanov-Lipshitz-Sarkar stable homotopy types for the
homotopical Khovanov homology of links in thickened surfaces indexed by moduli
space systems. This family includes the Khovanov-Lipshitz-Sarkar stable
homotopy type for the homotopical Khovanov homology of links in higher genus
surfaces (see the content of the paper for the definition). The question
whether different choices of moduli spaces lead to the same stable homotopy
type is open.
