I think if we can prove that the optimal coupling between uniform and comonotonic distribution is indeed given by $\pi$, then combining with your answer we can obtain a proof. I know one optimal coupling between uniform and comonotonic distribution is given by the monotone coupling which is different from $\pi$, but maybe due to the specialty of $\ell_1$-norm, $\pi$ is also an optimal coupling. I have tested on $N=4$ and they give the same value.