Fejes Tóth's zone covering conjecture

A zone is the spherical neighbourhood of a great circle, with width equal to twice its maximal spherical distance from that great circle. Let zones of widths $w_1,\ldots,w_n$ cover the unit disk. Fejes Tóth's zone covering conjecture. Their widths satisfy

$$\sum_{i=1}^n w_i\geq \pi.$$

The source presents this as Fejes Tóth's more general conjecture and says that the spherical conjecture discussed in the section was recently proven, so this claim is treated as solved.