The optimal partial plank covering conjecture for convex bodies

Let $K\restriction\mathbb{E}^d$ be a convex body and let $W>0$. A plank is a region between two parallel hyperplanes; its width is denoted by $w(P)$. For a finite family of planks $P_1,\ldots,P_n$, suppose that

$$\sum_{i=1}^n w(P_i)=W.$$

Optimal partial plank covering conjecture. There exists a plank $P$ of width $W$ such that, for every such finite family,

$$\operatorname{vol}_d\!\left(K\cap\bigcup_{i=1}^n P_i\right)\leq\operatorname{vol}_d(K\cap P).$$

This conjecture asserts that among all partial coverings of a convex body by planks with prescribed total width, a single plank is always optimal. It generalizes the affirmative result for Euclidean balls and the planar case described in the source, while the statement for arbitrary convex bodies and dimensions remains open.