The adjoint of the commutator on $\Omega J_{p^n-1}S^2$ is the Whitehead product $\Sigma (\Omega J_{p^n-1}S^2\wedge\Omega J_{p^n-1}S^2)\rightarrow J_{p^n-1}S^2$, which would vanish if $\Omega J_{p^n-1}S^2$ were homotopy commutative. This map is non-trivial since it is detected on the very bottom cell by a non-vanishing cup product. In fact on the bottom cell we see that it is the Hopf map $-2\eta:S^3\rightarrow S^2\hookrightarrow J_{p^n-1}S^2$ which generates $\pi_3S^2_{(p)}$.

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.