The union-of-balls simplex conjecture

For $N=n+1$, let $v_1,\ldots,v_N\in\mathbb R^n$ satisfy $|v_1|=\dots=|v_N|=r$, and let $B_{v_i}(R)$ denote the ball of radius $R$ centered at $v_i$. The union-of-balls simplex conjecture. For every $R>0$, the maximal volume of the union

$$\bigcup_i B_{v_i}(R)$$

is attained at any configuration forming a regular simplex inscribed into the ball of radius $r$. This is presented as a stronger statement that would imply the equal-radius simplex conjecture.