In https://arxiv.org/abs/1010.4252, Szabo defines a link invariant $\hat{H}(L)$ which can be computed combinatorially from a link diagram and shows that there is a spectral sequence from Khovanov homology to $\hat{H}(L)$. Conjecturally, this spectral sequence is isomorphic to the spectral sequence (proved by Ozsvath-Szabo) from Khovanov homology to $\hat{HF}(\Sigma(L))$ (the hat version Heegaard-Floer homology for the double branched cover). In particular, one should have $\hat{H}(L)\cong \hat{HF}(\Sigma(L))$.

To define $\hat{H}(L)$, Szabo divides the resolution diagrams into many types and writes down formula for the differential explicitly, case by case. While it is very concrete, I do not see how he obtains these formulas. I understand that he tries to add in differentials corresponding to higher dimensional faces of the resolution cube. But it seems that there are some deeper motivation to help him finding the right formula for these higher differentials. He does not explain much about these motivations in the paper.

My question is: Any one could explain the intuition underlying Szabo's construction?