Euclidean cylinder covering conjecture

Let $B^n$ be the Euclidean unit ball, and let $Z_1,\ldots,Z_N$ be $m$-dimensional Euclidean cylinders covering $B^n$. Write $\sigma_{n-m}(Z_i)$ for the $(n-m)$-dimensional volume of the cross-section of $Z_i$. The Euclidean cylinder covering conjecture. Then

$$\sum_{i=1}^N \sigma_{n-m}(Z_i)\geq \operatorname{vol}_{n-m}(B^{n-m})=\frac{\pi^{(n-m)/2}}{\Gamma((n-m)/2+1)}.$$

This restates a problem attributed in the source to Károly Bezdek and extends known special cases, including the plank case; the source does not state a resolution.