Hadwiger's illumination conjecture for convex bodies

Let $n\geq 3$ and let $K$ be an $n$-dimensional convex body. A positive homothetic copy of $K$ is a set of the form $x+\lambda K$ with $x\in\mathbb{R}^n$ and $0<\lambda<1$. Hadwiger's illumination conjecture. Every $n$-dimensional convex body can be covered by at most $2^n$ smaller positive homothetic copies of itself. Moreover, $2^n$ copies are required only when the body is an affine copy of the $n$-cube. The conjecture is a central covering formulation of the illumination problem; the supplied context describes the bound as widely believed but gives no resolution of the full statement.