Invariants of Heegaard splittings and trisections

Let $(\Sigma,\alpha,\beta,\gamma)$ be a trisection diagram for a closed 4-manifold $X$, in the sense of Gay--Kirby. After fixing a basepoint and the usual symplectic and almost complex data, one can define an $A_\infty$-algebra with underlying chain complex given by \[\mathcal{A} = \bigoplus_{i,j\in \{\alpha,\beta,\gamma\}} \widehat{\mathit{CF}}\,(i,j),\] where the multiplication maps count pseudo-holomorphic polygons in the standard way. In other words, $\mathcal{A}$ is the subcategory of the Fukaya category of $\mathrm{Sym^g}(\Sigma\setminus \{z\})$ generated by the Lagrangian tori $\mathbb{T}_\alpha, \mathbb{T}_\beta,\mathbb{T}_\gamma.$ Up to quasi-isomorphism of $A_\infty$-algebras, this should be invariant under isotopies and handleslides, and under changes to the other auxiliary data. In other words, the quasi-isomorphism type of $\mathcal{A}$ should be an invariant of the underlying trisection of $X$. The question below asks whether this is an interesting invariant:

Question: Can this construction distinguish non-diffeomorphic trisections of the same closed 4-manifold?

Note that $\mathcal{A}$ is not invariant under stabilization (even at the homology level), so it does not a priori give an invariant of closed 4-manifolds (which suggests that it might indeed be a good invariant of trisections!), but perhaps there is a way to extract a closed 4-manifold invariant from it:

Question: Can one extract an interesting closed 4-manifold invariant from this construction?

One should also ask these questions for the (completed) minus flavor of Heegaard Floer homology, and about bridge trisections for knotted surfaces in 4-manifolds. 

One can also ask similar questions for Heegaard diagrams, and perhaps this is the place to start. Given a Heegaard diagram $(\Sigma,\alpha,\beta)$ for a 3-manifold $Y$ and a choice of the usual auxiliary data, one can define an $A_\infty$-algebra \[\mathcal{B} = \bigoplus_{i,j\in \{\alpha,\beta\}} \widehat{\mathit{CF}}\,(i,j)\] in the same way as above. Its quasi-isomorphism type should likewise be an invariant of the underlying Heegaard splitting, leading to: 

Question: Can this construction distinguish non-homeomorphic Heegaard splittings of the same 3-manifold?

For a simple example on which to test this, one can study the non-homeomorphic genus-2 Heegaard splittings of the Brieskorn sphere $\Sigma(2,3,7)$ coming from the branched double covers of the standard bridge diagrams for the torus knot $T(3,7)$ and the pretzel knot $P(2,3,7)$. Are the $A_\infty$-algebras associated with these two splittings quasi-isomorphic?

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