Let it be some positive integer $n$, and some prime number $q<n$ such that $\gcd\left(q,n\right)=1$
Let it be $S=\left\{ x\,:\,0\leq x\leq n,\,\gcd\left(x,qn\right)=1\right\}$
I conjecture that $\mid S\mid>\lfloor\frac{\varphi\left(qn\right)}{q}\rfloor$, but following the bounding process I developed answering to the question Goldbach Conjecture: Subsets of the Euler Totient Function, the most I get for $n$ being some composite number is that $$\mid S\mid>\frac{\varphi\left(qn\right)+\omega\left(n\right)+2}{q}-\omega\left(n\right)$$
Is it the conjecture provable?
As posted in the answer to the question mentioned, it is easily provable for $n$ being some prime number, but not for $n$ being some composite number.
The conjecture follows applying the property that $\gcd\left(k,n\right)=\gcd\left(k+n,n\right)$, as this means that all numbers $k<qn$ such that $\gcd\left(k,qn\right)=1$ are somehow evenly distributed in the intervals $[1,n),[n+1,2n),...,[(q-1)n+1,qn)$. As there are $q$ intervals, one should expect to have approximately $\frac{\varphi\left(qn\right)}{q}$ elements of $S$ in each interval.
Thanks in advance for your help!