Bezdek–Zamfirescu's X-ray conjecture

Let $X(\mathbf{K})$ be the smallest number of lines such that every boundary point of a convex body $\mathbf{K}$ is X-rayed along at least one of them. X-ray conjecture. For every convex body $\mathbf{K}\subset\mathbb{E}^d$,

$$X(\mathbf{K})\leq 3\cdot 2^{d-2}.$$

The conjecture is proved in the plane but remains open in higher dimensions.