Does anyone know a good reference for a proof of the fact that given a dga $A$, an $A_\infty$-structure on $HA$ is ''the same'' as coherent choices for all of the higher Massey products of $HA$? More concretely the fact I am looking for is something like the following.

When defining the Massey product $\langle x_1,\dots, x_n\rangle$ there are multiple non-canonical choices that need to be made, which in turn give multiple cycles that could be called *the* Massey product of $x_1,\dots, x_n$. If $M(x_1,\dots, x_n)$ is the set of all possible resulting Massey products of $x_1,\dots, x_n$, and $\mu_n$ is the $n$-th $A_\infty$ structure map (on $HA$), then $$\mu_n(x_1\otimes\cdots\otimes x_n)\in M(x_1,\dots, x_n)$$
for all $n$ and $x_i$.