Kuperberg's packing–covering density conjecture for convex bodies

Let $d\geqslant 2$ be fixed. For a $d$-dimensional convex body $C$, let $\delta(C)$ and $\theta(C)$ denote its packing and covering densities by congruent copies. Kuperberg's conjecture. For every $\varepsilon>0$ there exists $\delta>0$ such that, for every $d$-dimensional convex body $C$, both

$$\delta(C)\leqslant 1-\varepsilon\implies\theta(C)\geqslant 1+\delta,$$

and

$$\theta(C)\geqslant 1+\varepsilon\implies\delta(C)\leqslant 1-\delta.$$

The conjecture says that, in fixed dimension, packing or covering density cannot approach the tessellation value $1$ independently of the other density. The paper refers to this as Kuperberg's conjecture and studies related quantitative bounds; its resolution status is not specified in the supplied text.