Suppose an iterated function system of similarity transformations $S_1, S_2, \dotsc, S_k:\mathbb{R}^n\to\mathbb{R}^n$ (with unique invariant set $F$) satisfies the open set condition for some non-empty bounded open set $O\subset \mathbb{R}^n$, so that $$\bigcup_{i=1}^k S_i(O)\subset O$$ with the union disjoint.
If $U\subset O$ is open and arbitrary, I would like to prove that $U$ is also a suitable choice in the OSC, so that $$\bigcup_{i=1}^k S_i(U)\subset U.$$
Clearly $\bigcup_{i=1}^k S_i(U)\subset \bigcup_{i=1}^k S_i(O)\subset O$, but is it the case that the setup necessarily implies $\bigcup_{i=1}^kS_i(U)\subset U$ with this union disjoint?
The statement is false. Consider the IFS on $\mathbb R$ consisting of the similarities $S_1(x)=x/2$ and $S_2(x)=x/2+1/2$. The open unit interval $(0,1)$ verifies OSC for this IFS. However, no interval of the form $(a,b)$ with $0<a<b<1$ will verify OSC.