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