Cap covering inradius conjecture

Let a spherical cap have radius $\alpha$, and let it be covered by a collection of convex spherical domains. For each domain, let its inradius be the radius of its largest inscribed spherical cap.

Cap covering inradius conjecture. The sum of the inradii of all domains in the collection is at least $\alpha$.

This problem is known for $\alpha\geq \pi/2$ but remains open for $\alpha<\pi/2$.