The Levi–Hadwiger covering conjecture for convex bodies

Let $K\subseteq\mathbb{R}^{n}$ be a convex body with nonempty interior, and let $N(K,L)$ denote the least number of translates of $L$ needed to cover $K$. A parallelotope is an affine image of an $n$-dimensional cube. Levi–Hadwiger covering conjecture. There exists $0<\lambda<1$ such that

$$N(K,\lambda K)\leq 2^{n}.$$

Moreover, equality holds if and only if $K$ is a parallelotope. Equivalently,

$$N(K,\operatorname{int}(K))\leq 2^{n}.$$

This is the classical covering problem for convex bodies by translates of slightly smaller homothetic copies. The source gives no resolution status; the equivalent formulations and the equality characterization are part of the stated conjecture.