Involutivity and ordinariness of the elementary integral manifold

Let $(M,\mathcal{I})$ be an analytic involutive exterior differential system, considered over $\mathbb{C}$. Write $\operatorname{elem}(\mathcal{I})$ for its elementary system, and let $D$ be the integral manifold associated to an ordinary integral manifold by the foliation theorem. Elementary involutivity conjecture. The system $\operatorname{elem}(\mathcal{I})$ is involutive on $M^{(1)}$, and the integral manifold $D$ is ordinary. This is presented as the strongest possible version of the preceding corollary, which establishes only that some prolongation of the elementary system is involutive; the conjecture concerns involutivity before prolongation and ordinariness of the resulting integral manifold.