Functoriality conjecture for the knot homology theory

Consider an embedded cobordism in $I\times S^3$ between knots in $S^3$, equipped with a Morse presentation and decomposed into Reidemeister moves and handle attachments. Assign maps to Reidemeister moves, births and deaths of circles, and saddle cobordisms using the indicated fixed isomorphisms and units or counits of the adjunctions.

Functoriality conjecture. This assignment of a map to a cobordism is independent of the choice of Morse function, i.e. this makes the knot homology theory $\mathcal{K}(-)$ functorial.

The conjecture would provide functoriality with respect to cobordisms, extending the known functorial behavior of Khovanov homology; the source presents it without a proof or resolution.