Spectral universality of sample covariance from unit sphere

Question

If $x_1,\dots,x_n\sim \mu$ are mean-zero iid samples in $R^d$ that are drawn from unit sphere, with covariance $E x x^\top = I_d,$ and $C_n:=\frac1n \sum_i^n x_i x_i^\top$ is the sample covariance, what can be said about spectral universality of $C_d$? Importantly, while the elements within each vector $x_i$'s are uncorrelated, they are not necessarily independent. this paper proves expected log-determinant of a Wishart matrix, which asymptotically becomes $E \log\det(C_d) \to -n^2/d$ as and $,dn\to \infty$ while $n/d < 1$. Under what assumption can this result be extended to $C_d$ and its log-determinant distribution?