Kepler's conjecture for asymptotic sphere-packing density

Let $\Gamma$ be a container, let $\rho(1,k\Gamma)$ denote the optimal density of unit-sphere packings in the $k$-fold magnification $k\Gamma$, and define

$$\hat{\rho}(\Gamma)=\limsup_{k\to\infty}\rho(1,k\Gamma).$$

For a large class of containers $\Gamma$, including those with piecewise smooth boundary, Kepler's conjecture.

$$\hat{\rho}(\Gamma)=\frac{\pi}{\sqrt{18}}.$$

This is the asymptotic formulation of the Kepler sphere-packing problem, asserting that sufficiently regular containers have the face-centered-cubic packing density. The supplied text does not indicate whether this formulation is intended as resolved or open.