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Getzler-Jones-Petrack introduced $A_\infty$ structures on the equivariantcomplex for manifold $M$ with smooth $\mathbb{S}^1$ action, motivated bygeometry of loop spaces. Applying Witten's deformation by Morse functionsfollowed by homological perturbation we obtained a new set of $A_\infty$structures. We extend and prove Fukaya's conjecture relating this Witten'sdeformed equivariant de Rham complexes, to a new Morse theoretical $A_\infty$complexes defined by counting gradient trees with jumping which are closelyrelated to the $\mathbb{S}^1$ equivariant symplectic cohomology proposed bySiedel.

Fukaya's conjecture on $S^1$-equivariant de Rham complex