Steiner symmetrization conjecture for the largest random angle

Let $K$ be a convex subset of ${\mathbb R}^d$, let $u$ be a unit vector in ${\mathbb R}^d$, and let $\mathsf{S}_uK$ denote the Steiner symmetrization of $K$ along $u$. For $n\geqslant3$, write $\varphi_{n,K}$ for the largest angle determined by $n$ random points in $K$. Steiner symmetrization conjecture. For every natural $d$, every natural $n\geqslant3$, every convex subset $K$ of ${\mathbb R}^d$, every unit vector $u$ in ${\mathbb R}^d$, and every $c\in(0,\pi)$, one has

$$\operatorname{\mathsf{P}}(\varphi_{n,\mathsf{S}_uK}>c)\leqslant\operatorname{\mathsf{P}}(\varphi_{n,K}>c).$$

Thus Steiner symmetrization is conjectured to make the largest angle stochastically smaller. The preceding proposition establishes that Steiner symmetrization does not increase the elongation coefficient, and the conjecture asks for the corresponding finite-sample stochastic comparison; its resolution is not indicated in the source.