Generalized affine plank conjecture for translative coverings

Let $C_1,\ldots,C_n\subset\mathbb{R}^d$ be convex sets. They permit a translative covering of a convex body $B\in\mathcal{K}^d$ if there exist vectors $x_1,\ldots,x_n\in\mathbb{R}^d$ such that

$$B\subset\bigcup_{i=1}^n(C_i+x_i).$$

Let $r_B(C_i)$ denote the relative inradius used in the affine plank formulation.

Generalized affine plank conjecture. Assume that the convex sets $C_1, \ldots, C_n \subset \mathbb{R}^d$ permit a translative covering of the convex body $B \in \mathcal{K}^d$. Then

$$\sum_{i=1}^n r_B(C_i)\geq 1$$

holds. The paper presents this as a generalization of the affine plank problem and proves it under special assumptions; the supplied text does not establish the conjecture in full.