A Steenrod square on Khovanov homology and a cup-i product

Lipshitz-Sarkar defined a stable homotopy type refining Khovanov homology, producing cohomology operations ${\text{Sq}}^i$ on the Khovanov homology $Kh(L)$ of a link $L$. Later, Morán proposed a sequence of cup-i products on the ${\mathbb{F}}_2$-coefficient cochain complex of any augmented semi-simplicial object in the Burnside category. Applied to the Khovanov functor, he obtained another sequence of operations ${\mathfrak{s}\mathfrak{q}}^n$ on $Kh(L)$, where ${\mathfrak{s}\mathfrak{q}}^0$, ${\mathfrak{s}\mathfrak{q}}^1$ agree with the usual Steenrod squares. We prove that Lipshitz-Sarkar’s ${\text{Sq}}^2$, the first Steenrod operation that cannot be computed from merely homological data, agrees with Morán’s ${\mathfrak{s}\mathfrak{q}}^2$.