Bezdek–Schneider's spherical inradius covering conjecture

Let a cap on the unit sphere have spherical radius $r$. A convex spherical domain is a domain in which any two points can be joined by an arc of a great circle lying in the domain; its inradius is the spherical radius of the largest cap it contains. Suppose the cap is covered by a collection of convex spherical domains.

Bezdek–Schneider's conjecture. The total inradius of the domains is at least $r$.

The conjecture is attributed to Bezdek and Schneider. The supplied text says it was solved for $r\geq\pi/2$, so the general statement remains open.