The uniqueness of homotopy associative, homotopy commutative multiplications on certain H-spaces

Let p be an odd prime. We show that for a certain family of p-local finite H-spaces, if a homotopy associative, homotopy commutative multiplication exists then it is unique. In particular, this answers a long-standing question of whether a homotopy associative, homotopy commutative multiplication on S2n+1 is unique.