Bang's affine plank conjecture

Let $C$ be a convex body, and let $P_1,\ldots,P_n$ be planks covering it. If $H_i$ is parallel to the boundary hyperplanes of $P_i$, define the relative width of $P_i$ by $w(P_i)/w(C,H_i)$, where $w(C,H_i)$ is the width of $C$ in the direction of $H_i$. Bang's affine plank conjecture. The sum of the relative widths is at least $1$:

$$\sum_{i=1}^n \frac{w(P_i)}{w(C,H_i)}\geq 1.$$

This affine-invariant strengthening of Tarski's plank problem is explicitly described later in the survey as an important open problem in convex geometry.