This is the subject of Section 2 in that paper (sorry for self-promotion). Chains actually have two $B_\infty$-module structures (over cochains). They are moreover compatible (see Theorem 2.4 of the above paper).

This is the subject of Section 2 in that paper (sorry for self-promotion). Chains actually have two $B_\infty$-module structures (over cochains). Those two module structures are moreover compatible (see Theorem 2.4 of the above paper for a precise statement).