Equality of the characteristic spans of the restricted and elementary varieties

Let $(M,\mathcal{I})$ be an involutive exterior differential system. Let $\dot{\Xi}$ and $\Xi_E$ be the characteristic varieties defined in the surrounding discussion, viewed as subsets of $X^*$. Characteristic-span conjecture. Their linear spans satisfy

$$\langle\dot{\Xi}\rangle=\langle\Xi_E\rangle$$

as subspaces of $X^*$. This is a weaker version of the preceding characteristic-sheaf conjecture and is stated as sufficient for computing $\operatorname{elem}^2(\mathcal{I})$, even without establishing its involutivity.