The spherical Goodman covering conjecture for caps

Let $r_1,\dots,r_n$ be the spherical radii of caps lying on a hemisphere of the unit $d$-sphere. Assume that no great sphere divides the caps into two non-empty sets without intersecting any cap.

Spherical Goodman conjecture. The given caps can be covered by a cap of spherical radius

$$r=\sum_{i=1}^n r_i.$$

This is proposed as the spherical relative of the cited Goodman theorem and remains open in the supplied text. The hemisphere assumption is noted there to be indispensable.