Maximal-order uniqueness conjecture for maximal doubly even codes

Let $C\subset \mathbb{F}_2^n$ be a maximal doubly even binary code, and let $I=(i_0,i_1,\ldots,i_{n-1})$. Consider the Clifford algebra $\mathbb{C}_n$ and the rational Clifford algebra $\left(\frac{(-1)^{n-1}}{\mathbb{Q}}\right)$. Maximal-order uniqueness conjecture. For every maximal doubly even binary code $C\subset \mathbb{F}_2^n$, there exists a unique maximal order $\mathcal{O}_C\subset \left(\frac{(-1)^{n-1}}{\mathbb{Q}}\right)\subset \mathbb{C}_n$ containing the Clifford order such that

$$\mathcal{O}_C \supset \mathbb{Z}\left[\frac{c\cdot I}{2}:c\in C\right].$$

The assertion is false: the Hurwitz order provides a counterexample to the related claim that every order equals the algebra generated by its vector part, so the proposed uniqueness statement does not hold in the stated generality.