Cube-compatible identification conjecture for HOMFLY-PT and knot Floer homologies

Let $D$ be a braid diagram for a knot $K$, and let $S$ be a singular resolution of $D$. Define $H_H(S)$ to be the HOMFLY-PT homology of $S$, and define $H_F(S)$ to be the homology of the complex assigned to $S$ by the knot Floer cube of resolutions.

Cube-compatible identification conjecture.

There is an isomorphism

$$H_F(S)\cong H_H(S).$$

Moreover, this isomorphism commutes with the induced edge maps in the cube of resolutions.

Such an identification would explain the relationship between the HOMFLY-PT and knot Floer cube constructions at the level of individual singular resolutions and their transition maps. The supplied context does not state whether this conjecture has been resolved.