Separation Conditions for Iterated Function Systems with a Common Attractor

Let $K$ be the non-singleton attractor of a finite iterated function system $Φ=\{ϕ_i(x)=a_i O_i x+t_i\}_{i=1}^m$ on $\mathbb{R}^d$, where $0<a_i<1$, $t_i\in\mathbb{R}^d$ and the $O_i$ are orthogonal transformations on $\mathbb{R}^d$. Suppose that $Φ$ satisfies the open set condition, but not the str