At first sight, the Fukaya category has obvious cyclic symmetry, because the $A_infty$ structure maps count points in spaces of rigid pseudo-holomorphic polygons subject to Lagrangian boundary conditions, and these spaces depend only on the cyclic order of the Lagrangians. This indeed proves that the cohomological Fukaya category, in which the hom-spaces are Floer cohomology spaces, is cyclically symmetric.
The trouble comes when these Lagrangians don't intersect one another transversely - for instance, the same Lagrangian occurs more than once - because then the morphism spaces and structure maps invoke Hamiltonian perturbations which need not be cyclically symmetric. The problem which Fukaya has solved over the reals (see Matthew Ballard's answer) is, I presume, to find a way to make these perturbations cyclically symmetric whilst also achieving the necessary coherence between them, as well as transversality for compactified moduli spaces of inhomogeneous pseudo-holomorphic polygons (or worse, their abstract perturbations). These are the things which actually define the $A_infty$-structure.
FOOO worked extremely hard to get geometrically-meaningful units in their Fukaya endomorphism algebras, where other authors are content to define units by tweaking the $A_infty$-structure algebraically. My hope would be that algebra will also give a cheaper approach to cyclic symmetry, particularly since I'm told that for Costello's theorem to hold, one only needs "derived" cyclic symmetry.
By the way, let's be clear that Costello's theorem, suggestive as it may be, is not about GW invariants! It's about theories over $M_{g,n}$, not over $overline{M}_{g,n}$.
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