Linnik's covering-radius conjecture for projected lattice points

Let $\mathcal E(n)$ be the projected lattice points on $S^2$, and let $N_n$ denote their number. The covering radius $M(P_1,\dots,P_N)$ of a configuration on $S^2$ is the least $r>0$ such that every point of $S^2$ lies within distance at most $r$ of some $P_j$. Linnik's covering-radius conjecture.

$$M(\mathcal E(n))=N_n^{-1/2+o(1)}\quad\text{as }n\to\infty.$$

An area argument gives the lower bound $M\geq4/\sqrt N$, while effective equidistribution currently yields only a weaker power bound. The conjecture is presented as a consequence of Linnik's shrinking-set conjecture and remains open.