co.combinatorics – Happy ants never leave compact domain?

I am curious if the following seemingly simple question has an easy answer?

Consider an ant population of $N$ ants that lives in $mathbb R^2$. Each ant can be considered to be a disk or radius $1.$ Ants like to be close to their peers but also not too close. The optimal distance between the center of two ants is $5^{2/3}$. So given two coordinates of centers of ants $x_i,x_j in mathbb R^2$. Their happiness $H$ is $H(x_i,x_j):=-vert vert x_i-x_j vert^{3/2} -5 vert.$ Distances $vert x_i-x_jvert le 1$ are not allowed since ants cannot be on top of each other.

Now consider the total happiness $H_N:=sum_{i<j} H(x_i,x_j).$ The question is the following:

Consider any maximiser $x=(x_1,..,x_N)$ of happiness $H_N.$ Can we show that any optimizer is always contained in a ball $B(y(x),rsqrt{N})$ where $r$ is a universal positive constant independent of $N$ and the maximiser and $y$ is allowed to depend on $x$. That means, all $x_i$ of the optimizer are in $B(y(x),rsqrt{N}).$

The conjecture of $B(y(x),rsqrt{N})$ is due to the fact that order $N$ lattice particles would fit into a ball of radius $B(y(x),rsqrt{N}).$

The question sounds almost like a school problem, but I just cannot exclude weird configurations.