James proved the homotopy decomposition $\Sigma\Omega\Sigma X\simeq \bigvee_{n=1}^\infty \Sigma X^{\wedge n}$. This is a natural homotopy equivalence for a pointed connected CW complex $X$. Here $X^{\wedge n}$ is the $n$-fold self-smash product of $X$.

Is there a counterexample to the stronger assertion that $\Omega\Sigma X\simeq \bigvee_{n=1}^\infty X^{\wedge n}$? This assertion implies James' decomposition as suspension, being left adjoint to looping, commutes with wedge sum.