Canonicity conjecture for the Lee functor on framed link cobordisms

Let ${\mathcal Cob}_{f/i}^4$ be the quotient of ${\mathcal Cob}_f^4$ by framed movie moves. Assume the chain transformations in the Lee functor construction are made canonical, independently of sign choices. Canonicity conjecture. The functor

$$\mathcal{F}_{\operatorname{Lee}}\circ\operatorname{Kh}_{1,1}:{\mathcal Cob}_{f/i}^4\rightarrow \operatorname{Kom}_{/\pm h}(\operatorname{Kom}_{/h}(\mathbb{Z}\text{-}\operatorname{mod}))$$

descends to the quotient by framed movie moves. This would make the Lee-theoretic Khovanov construction invariant under the relevant movie moves, although the source does not establish the required independence of sign choices or the descent.