Leinster–Willerton magnitude conjecture for convex bodies

Let $X\subset\mathbb{R}^n$ be compact and convex, and let $\mathcal{M}_X(R):=\operatorname{mag}(X,R\cdot\mathrm{d})$ for $R>0$. For $0\leq k\leq n$, let $V_k(X)$ denote the intrinsic volumes of $X$, let $\omega_k$ be the volume of the $k$-dimensional unit ball, and let $\operatorname{vol}_j$ denote $j$-dimensional volume. Leinster–Willerton magnitude conjecture. One should have

$$\mathcal{M}_X(R)=\frac{1}{n!\omega_n}\operatorname{vol}_n(X)R^n+\frac{1}{2(n-1)!\omega_{n-1}}\operatorname{vol}_{n-1}(\partial X)R^{n-1}+\cdots+1$$

with, equivalently,

$$\mathcal{M}_X(R)=\sum_{k=0}^n\frac{1}{k!\omega_k}V_k(X)R^k.$$

Here $\omega_k$ is the volume of the $k$-dimensional unit ball. This conjecture predicts that magnitude recovers the intrinsic volumes of a compact convex body, and in particular that its constant expansion coefficient agrees with the Euler characteristic. The general statement is presented as a long-standing problem, especially for compact domains in even-dimensional Euclidean spaces; the supplied text does not establish its resolution.