Mutation-invariance of Khovanov-Floer theories

Khovanov-Floer theories are a class of homological link invariants which admit spectral sequences from Khovanov homology. They include Khovanov homology, Szab{\'o}'s geometric link homology, singular instanton homology, and various Floer theories applied to branched double covers. In this short note we show that certain strong Khovanov-Floer theories, including Szab{\'o} homology and singular instanton homology, are invariant under Conway mutation. This confirms conjectures of Seed and Lambert-Cole. Along the way we prove two other conjectures about the structure of Szab{\'o} homology.