$d$-ball approximation for $d\gg 1$ with a convex hull of random points on its boundary

Given a $d$-ball $\mathcal{S}^{d}$, let $P_n$ a set of $n$ points selected uniformly at random on the boundary $\mathcal{S}^{d-1}$ of $\mathcal{S}^{d}$. Let $\mathcal{C}_n$ the convex hull of $P_n$. We denote by $V(\mathcal{S}^{d})$ and $V(\mathcal{C}_n)$ the volume of $\mathcal{S}^d$ and $\mathcal{C}_n$ respectively.

Question: Can we prove that, for all constant values $a\in (0,1)$, $n$ has to grow superpolynomially in $d$ (for $d\to\infty$) to satisfy $\mathbb{E}[V(\mathcal{C}_n)]\ge aV(\mathcal{S}^{d})$ ?

Note:Do you please have any reference for the convex hull $\mathcal{C}_n^*$ of the $n$ points of $\mathcal{S}^{d-1}$ maximizing $V(\mathcal{C}_n)$ (so that we can hopefully avoid to calculate the expectation $\mathbb{E}[V(\mathcal{C}_n)]$ if $n=\omega(\mathrm{poly}(d))$ to satisfy $V(\mathcal{C}_n^*)\ge aV(\mathcal{S}^{d})$ for any $a\in(0,1)$ when $d\to\infty$) ?