On Steenrod squares for even and odd Khovanov homology

For an arbitrary link $L\subset S^3$ , Sarkar-Scaduto-Stoffregen construct a family of spatial refinements of even and odd Khovanov homology. We give a computation of ${\text{Sq}}^2$ on these spaces, determining their stable homotopy types for all knots K up to 11 crossings. We also prove that the Steenrod squares ${\text{Sq}}_0^2$ , ${\text{Sq}}_1^2$ defined by Schütz do arise as Steenrod squares on these spaces.