Bezdek–Bezdek relative-width conjecture

Let ${\bf C}$ and ${\bf K}$ be convex bodies in ${\mathbb E}^d$. For a plank ${\bf P}$ parallel to a hyperplane $H$, define its ${\bf C}$-width by

$${\rm w}_{\bf C}({\bf P})=\frac{{\rm w}({\bf P})}{{\rm w}({\bf C},H)},$$

and define the minimal ${\bf C}$-width of ${\bf K}$ by

$${\rm w}_{\bf C}({\bf K})=\min_H\frac{{\rm w}({\bf K},H)}{{\rm w}({\bf C},H)}.$$

Bezdek–Bezdek relative-width conjecture. If ${\bf K}$ is covered by planks ${\bf P}_1,\dots,{\bf P}_n$ in ${\mathbb E}^d$, then

$$\sum_{i=1}^n {\rm w}_{\bf C}({\bf P}_i)\ge {\rm w}_{\bf C}({\bf K})$$

for any convex body ${\bf C}$ in ${\mathbb E}^d$. The source presents this as an equivalent version of Bang's strengthened plank conjecture; its resolution is not given in the supplied text.