Closed formula for the flip-sign action on link homology

Derive a closed formula for the sign of the flip map on Khovanov–Rozansky $gl_N$ link homology, accounting for the passage from the foam module $V_\Gamma$ to the state space $\langle\Gamma\rangle$ and then from the chain complex to link homology.

Background

The paper proves that the flip map on the chain complex associated to an uncolored equivariant Khovanov–Rozansky $gl_N$ link diagram is diagonalizable with eigenvalues $\pm1$. For planar webs, Proposition 5.5 determines the sign on generators represented by capped-off MOY foams through the quantity $(-1)^{\chi_2^-(W)}$.

The author explains that extending this generator-level description to a closed formula on link homology is unresolved because one must first choose a distinguished eigenbasis when passing from the foam module $V_\Gamma$ to the state space, and then compute homology when passing from the chain complex to link homology. A formula at the homology level could clarify the resulting $\pm1$-eigenspaces and their possible topological significance.