Let $L$ be a link in $\mathbb{R}^3$, with $D, D'$ be diagrams in $\mathbb{R}^2$ representing $L$. Khovanov constructed two graded chain complexes $$C_{D} = (Ch_{D}, d_{D}) \quad C_{D'}=(Ch_{D'}, d_{D'})$$ and proved that their homologies $Kh(D), Kh(D')$ are isomorphic. This defines an invariant for the link $L$, namely the Khovanov homology $Kh(L)$.
Mantra in algebraic topology suggests that we should not be taking homology too quickly (to avoid forgetting useful information).
I thus wonder if there are some weaker equivalences between $C_{D}$ and $C_{D'}$. Indeed there is: the Khovanov homology is known to be a degeneration of Khovanov homotopy type (i.e. a refined invariant $\tilde{Kh}(L))$ which is a homotopy type up to homotopy equivalence). How about a version in terms of equivalence of $A_\infty$ algebras? If such Khovanov $A_\infty$ algebra indeed exists, is it stricly weaker than the Khovanov homotopy type?
