Smurov's plank covering conjecture for the unit ball

Let $d$ be a positive integer, and let a plank in $\mathbb R^d$ be the region between two parallel hyperplanes; its width is the distance between those hyperplanes. A collection of planks has total width equal to the sum of their widths.

Smurov's conjecture. There exists a constant $C_d$, possibly depending on $d$, such that any collection of planks in $\mathbb R^d$ with total width at least $C_d$ can be translated so that the translated planks cover the unit ball.

The source attributes this conjecture to Mikhail Smurov and records that $C_2<2+\pi$ is known, while the case $d>2$ remains open.