Mirror-duality conjecture for the two second Steenrod squares

Let $L$ be a link and let $\overline{L}$ be its mirror. For $\varepsilon\in \mathbb{Z}/2\mathbb{Z}$, the Khovanov cohomology groups of $L$ and the dual groups of its mirror are related by the natural isomorphisms displayed in the diagram, with the second Steenrod square $\operatorname{Sq}^2_\varepsilon$ on the left and the dual of $\operatorname{Sq}^2_{1+\varepsilon}$ on the right.

Mirror-duality conjecture. The diagram

\begin{tikzpicture} \node at (0,2) {$\operatorname{Kh}^{i,j}(L;\mathbb{Z}/2\mathbb{Z})$}; \node at (5.6,2) {$\operatorname{Hom}(\operatorname{Kh}^{-i,-j}(\overline{L};\mathbb{Z}/2\mathbb{Z}),\mathbb{Z}/2\mathbb{Z})$}; \node at (0,0) {$\operatorname{Kh}^{i+2,j}(L;\mathbb{Z}/2\mathbb{Z})$}; \node at (5.6,0) {$\operatorname{Hom}(\operatorname{Kh}^{-i-2,-j}(\overline{L};\mathbb{Z}/2\mathbb{Z}),\mathbb{Z}/2\mathbb{Z})$}; \draw[->] (1.35,2) -- node[above] {$\cong$} (3,2); \draw[->] (1.5,0) -- node[above] {$\cong$} (2.8,0); \draw[->] (0,1.7) -- node[\right] {$\operatorname{Sq}^2_\varepsilon$} (0,0.3); \draw[->] (5.6,1.7) -- node[\right] {$(\operatorname{Sq}^2_{1+\varepsilon})^\ast$} (5.6,0.3); \end{tikzpicture}

commutes. Here the right vertical map is the dual of the second Steenrod square indexed by $1+\varepsilon$.

This conjecture proposes the Spanier--Whitehead duality behavior for the two second Steenrod squares, extending the known duality relation for the Lipshitz--Sarkar stable homotopy type. The claimed behavior was confirmed by calculations for links in the Rolfsen table, but its validity for arbitrary links remains open.