Conjecture on the covering radius of random points on the sphere

Let $d\geq 2$, let $X_N$ be a configuration of $N$ i.i.d. uniformly distributed points on $\mathbb{S}^{d}$, and let $\rho(X_N)$ denote its Euclidean covering radius. Define $B_d^\prime=\kappa_d^{-1/d}$.

Covering-radius conjecture. The expected covering radius satisfies

$$\mathbb{E}[\rho(X_N)]\sim B_d^\prime\left(\frac{\log N}{N}\right)^{1/d}\qquad\text{as $N\to\infty$}.$$

This predicts the logarithmic correction governing the largest hole among random points on the sphere. It is motivated by the conjectured limiting law for individual scaled hole radii and concerns the asymptotic expectation of the largest hole.