The magnitude formula for compact ell_1-convex sets

Let $A$ be a compact $\ell_1$-convex set, meaning a geodesic subset of $\ell_1^n$, and let $V'_i(A)$ denote its $i$th $\ell_1$-intrinsic volume. These volumes are defined by the Steiner-type formula

$$\operatorname{vol}(A+r\mathcal{C}^{n})=\sum_{i=0}^n V'_i(A)r^{n-i},$$

where $\mathcal{C}^{n}=[-1/2,1/2]^n$. Magnitude formula. For all compact $\ell_1$-convex sets $A\subseteq\ell_1^n$,

$$\left| A \right| = \sum_{i=0}^n \frac{V'_i(A)}{2^i}.$$

The formula extends the corresponding result for boxes and would identify magnitude with a weighted sum of the $\ell_1$-intrinsic volumes. The supplied excerpt presents it as the conjectural generalization beyond boxes; its resolution is not indicated.