Two second Steenrod squares for odd Khovanov homology

Recently, Sarkar–Scaduto–Stoffregen constructed a stable homotopy type for odd Khovanov homology, hence obtaining an action of the Steenrod algebra on Khovanov homology with $\mathbb Z/2\mathbb Z$ coefficients. Motivated by their construction we propose a way to compute the second Steenrod square. Our construction is not unique, but we can show it to be a link invariant which gives rise to a refinement of the Rasmussen $s$-invariant with $\mathbb Z/2\mathbb Z$ coefficients. We expect it to be related to the second Steenrod square arising from the Sarkar–Scaduto–Stoffregen construction.