Do higher order $A_{\infty}$ multiplications defined by Fukaya vanish when Morse cochain complex is isomorphic to Morse cohomology, in which case the cup product is associative, e.g. $S^n, \mathbb{C}P^n$ with the standard Morse funcitons on them?

Can the higher order $A_{\infty}$ multiplications defined by Fukaya be made trivial(by perturbing gradient trees) when Morse cochain complex is isomorphic to Morse cohomology, in which case the cup product is associative, e.g. $S^n, \mathbb{C}P^n$ with the standard Morse funcitons on them?