Strong Hall's conjecture for triangle-shape distributions

Let $S$ be a convex region, let $F_S(\phi)$ denote the distribution function for the shape of a random triangle formed by three points chosen from $S$, and let $F_{n-\rm ball}(\phi)$ denote the corresponding distribution function for a ball of the same normalized volume. Strong Hall's conjecture. For every relevant shape parameter $\phi$,

$$F_S(\phi) \leq F_{n-\rm ball}(\phi).$$

Hall's original conjecture is the special case at $\phi=\frac{\pi}{2}$, where Hall's notation is $P(S)=F_S(\frac{\pi}{2})$. The source introduces this as a natural generalization of Hall's conjecture, but supplies no resolution here.