Leinster–Willerton planar magnitude conjecture

Let $X\subseteq\mathbb{R}^2$ be a compact set suitable for the perimeter and classical Euler characteristic to be defined. Here $|tX|$ denotes the magnitude of the dilation of $X$ by the factor $t$, and ${\rm Area}(X)$, ${\rm Perim}(X)$, and $\chi(X)$ denote its area, perimeter, and classical Euler characteristic. Leinster–Willerton's conjecture. As $t\to\infty$,

$$|tX|=\frac{1}{2\pi}{\rm Area}(X)t^2+\frac{1}{4}{\rm Perim}(X)t+\chi(X)+o(1).$$

The conjecture predicts that the asymptotic magnitude of a suitable planar compact set records area, perimeter, and Euler characteristic. The supplied context presents numerical and parallel metric evidence, but gives no resolution.