Zong's blocking-number conjecture

Let $\beta(\mathbf{K})$ denote the blocking number of a $d$-dimensional convex body $\mathbf{K}$. Zong's blocking-number conjecture. For every $d$-dimensional convex body $\mathbf{K}$,

$$2d\leq\beta(\mathbf{K})\leq 2^d,$$

and $\beta(\mathbf{K})=2^d$ if and only if $\mathbf{K}$ is a $d$-dimensional cube. The paper presents this as an open conjecture concerning the number of mutually nonoverlapping translates needed to block further translates from touching the body.