Khovanov's presentation-independence conjecture for link-cobordism maps

Let $S$ be a compact oriented surface and let $J,\widetilde J$ be representations of $S$ by sequences of link diagrams, with endpoint diagrams $J_0,J_1$ and $\widetilde J_0,\widetilde J_1$. Let $\theta_J$ and $\theta_{\widetilde J}$ be the induced maps on cohomology. Khovanov's presentation-independence conjecture. If $J_0$ and $\widetilde J_0$ are isomorphic and $J_1$ and $\widetilde J_1$ are isomorphic, then

$$\theta_J=\pm\theta_{\widetilde J}.$$

This is the precise movie-presentation form of link-cobordism invariance and would make the induced maps depend only on the isotopy class of the surface, up to sign.