This was answered in the affirmative by Brayton Gray in his paper *Homotopy Commutativity and the EHP Sequence*. Specifically he shows that for all $n$ the space $\Omega J_{p^s-1} S^{2n}$ is homotopy commutative for $s\geq 1$ when localised at any prime $p\geq 3$. Moreover he claims to be able to show that $\Omega J_{jp^s-1}S^{2n}$ is homotopy commmutative for $s\geq 1$ and $j\leq p$ odd, although he does not give a full proof.

In the same paper he also obtains results on the homotopy commutativitivy of the classifying space $B_{2n-1,r}$ of the iterated suspension.