The 3-primary classifying space of the fiber of the double suspension

Gray showed that the homotopy fiber W-n of the double suspension S2n-1 <(E-2)under right arrow> Omega S-2(2n+1) has an integral classifying space BWn, which fits in a homotopy fibration S2n-1 <(E-2)under right arrow> Omega S-2(2n+1) <(nu)under right arrow>BWn. In addition, after localizing at an odd prime p, BWn is an H- space and if p >= 5, then BWn is homotopy associative and homotopy commutative, and nu is an H-map. We positively resolve a conjecture of Gray's that the same multiplicative properties hold for p = 3 as well. We go on to give some exponent consequences.