One-sided concentration conjecture for random spherical coverings

Let $\epsilon=\epsilon(n)$ be the radius of the geodesic balls, let $N_{c,\epsilon(n)}$ be the number of independently and uniformly selected centers used for the covering, and let $g(\mathbf X)$ be the covered surface measure. One-sided concentration conjecture. For every $\epsilon>0$,

$$\mathbb P\mathopen{}\mathclose\bgroup\originalleft(g(\mathbf X)-(1-e^{c})>\epsilon\aftergroup\egroup\originalright)\leq C e^{-c(n)N_{c,\epsilon(n)}\epsilon^2},$$

where $c(n)\to\infty$ as $n\to\infty$. This is intended to sharpen the existing two-sided concentration estimate when the geodesic-ball radius is sufficiently small.