Uniqueness of the pair-of-pants A-infinity structure

Let $\Pi_n$ be the $n$-dimensional pair of pants, let $L_I$ be the relevant objects, and consider the algebra $\bigoplus_{I,J}HW^*(L_I,L_J)$. Assume its $A_\infty$-structure is compatible with the grading, every output generator of a higher product represents in $H_1(\Pi_n,\mathbb{Z})$ the sum of the input classes, and

$$\mu^3(u_I,u_J,u_K)=\pm\operatorname{id}$$

for all disjoint $I,J,K$ with $I\sqcup J\sqcup K=\{0,\dots,n+1\}$. Pair-of-pants $A_\infty$-uniqueness conjecture. Up to homotopy, there is a unique such $A_\infty$-structure. The proposed uniqueness would reduce the higher products to the grading, relative-homology condition, and the specified ternary products. The source describes this as a conjectural algebraic statement and does not give a general proof.