The spherical-cap simplex conjecture

Let $\mathbb S^{n-1}$ be the unit sphere, and consider $N=n+1$ spherical caps of radius $R$ on it. The spherical-cap simplex conjecture. For every $R>0$, the maximal area of the union of these $N$ caps is attained when their centers form a regular simplex inscribed into the unit sphere. This is presented as an even stronger statement implying the union-of-balls conjecture.