Functoriality of knot Floer homology under marked smooth surface isotopy

Let $L_0$ and $L_1$ be oriented links represented by grid diagrams, and let a sequence of grid diagram maps represent a grid cobordism between them. These maps induce maps on the knot Floer complexes and hence on $\operatorname{HFK}^-$.

Grid-cobordism functoriality conjecture. Grid diagram maps induce maps on $\operatorname{HFK}^-$ that are invariant with respect to marked smooth isotopy classes of surfaces.

The conjecture seeks a grid-diagram formulation of the functoriality of knot Floer homology under smooth decorated cobordisms. At this stage, the correspondence between smooth surfaces and sequences of grid moves, as well as the relation between smooth surface isotopies and grid movie isotopies, was not yet understood; marking issues were addressed in subsequent work.