Doubly even code order conjecture

Let $C$ be a doubly even binary code of length $n$, and let $\Lambda_C$ be the inverse image of $C$ under the natural map

$$\mathbb{Z}+\mathbb{Z}i_1+\cdots+\mathbb{Z}i_{n-1}\longrightarrow\mathbb{F}_2+\mathbb{F}_2i_1+\cdots+\mathbb{F}_2i_{n-1}.$$

Let $\mathbb{Z}[\frac{1}{2}\Lambda_C]$ denote the ring generated by the half-lattice. Doubly even code order conjecture. The ring $\mathbb{Z}[\frac{1}{2}\Lambda_C]$ is an order in the rational Clifford algebra $\left(\frac{(-1)^{n-1}}{\mathbb{Q}}\right)$. This is part of the open correspondence between lattices and doubly even codes; the source states that related problems remain outstanding.