Inequality for sum of sines over integers in $[1;x]$

Question

It's easy to prove that $\sum_\limits{n=1}^x|\sin(n)|\sim\frac{2x}{\pi}$, using equdistribution of $\{n\;(mod\;\pi)\}$ on $[0;1]$. Define $S(x)=\sum_\limits{n=1}^x\left(|\sin(n)|-\frac{2}{\pi}\right)$. Say, $S=\sup_\limits{x\in\mathbb{N}}|S(x)|$. Is it true that $S\leqslant1$?