Extremal beta-polytope intrinsic-volume limit law conjecture

Let $U_1,\ldots,U_N$ be independent random vectors in $\mathbb{R}^d$ with beta density

$$p_{d,\beta}(x)=c_{d,\beta}(1-\\|x\\|^2)^\beta\mathbf{1}_{\mathbb B^d}(x),$$

where $\beta>-1$, $\mathbb B^d=\\{x\in\mathbb{R}^d:\\|x\\|\leq 1\\}$, and $v_m$ denotes the $m$-th intrinsic volume. Extremal beta-polytope conjecture. For any fixed $d,n\in\mathbb N$, $\beta>-1$, and $m\in\\{0,1,\ldots,d\\}$, there exist positive numbers $A=A(d,m,n,\beta)$, $B=B(d,m,n,\beta)$, and $C=C(d,m,n,\beta)$ such that, for every $t$,

$$\lim_{N\to\infty}\mathbb{P}\left[N^A\left(\max_{x_1,\ldots,x_n\in\mathbb B^d}v_m([x_1,\ldots,x_n])-\max_{1\leq i_1<\cdots<i_n\leq N}v_m([U_{i_1},\ldots,U_{i_n}])\right)\leq t\right]=1-e^{-Bt^C}.$$

This conjectures a Weibull-type limit law, with suitable positive scaling and distribution parameters, for the deficit between the optimal intrinsic volume among all $n$-vertex polytopes in the unit ball and the largest intrinsic volume among the random beta polytopes generated by $N$ sampled points.