Functoriality conjecture for marked link Floer homology

Let $L_i^\bullet=(L_i,\boldsymbol{p}_i)$ be marked links, and let a marked surface be a smoothly embedded surface in four-space with boundary $L_1^\bullet\cup L_2^\bullet$.

Functoriality conjecture. Link Floer homology is functorial with respect to marked smooth isotopy classes of surfaces embedded in four-space: a marked surface with boundary $L_1^\bullet\cup L_2^\bullet$ induces a map

$$\operatorname*{HFL}(L_1^\bullet)\to\operatorname*{HFL}(L_2^\bullet),$$

and two different marked surfaces that are isotopic induce identical maps.

This conjecture seeks a marked-surface analogue of the functoriality of Khovanov homology under smooth isotopy classes of orientable surfaces in four-space, addressing the role of markings in making link Floer homology maps well defined.