 Solution

It is not uniform. For a small fixed $c>0$, set $x_N=c/N^3$. On a block $N\le n\le (1+\delta )N$, all $n^3{x}_N$ lie in a compact subinterval of $(0,\pi )$, so the corresponding tail has a positive lower bound independent of $N$; this violates the uniform Cauchy criterion.