The spherical-segment covering conjecture for the unit sphere

A spherical segment is the solid obtained by cutting the unit ball with a pair of parallel planes, with width measured as the length of the shortest arc on the sphere whose endpoints touch both parallel planes. Consider a collection of spherical segments covering the unit sphere.

Spherical-segment covering conjecture. The total width of the spherical segments is at least $\pi$.

This is presented as an even stronger version of the preceding unit-ball conjecture and remains open in the supplied text.