KöMaL 2022 A823

For positive integers $n$ consider the lattice points $S_n=\big\{(x,y,z)\colon 1\le x\le n$, $1\le y\le n$, $1\le z\le n$, $x,y,z\in \mathbb{N}\big\}$. Is it possible to find a positive integer $n$ for which it is possible to choose more than $n\sqrt{n}$ lattice points from $S_n$ such that for any two chosen lattice points at least two of the coordinates of one is strictly greater than the corresponding coordinates of the other?