Finer geometric rigidity of limit sets of conformal IFS

We consider infinite conformal iterated function systems in the phase space $\mathbb{R}^d$ with $d\ge 3$. Let $J$ be the limit set of such a system. Under a mild technical assumption, which is always satisfied if the system is finite, we prove that either the Hausdorff dimension of $J$ exceeds the topological dimension $k$of the closure of $J$ or else the closure of $J$ is a proper compact subset of either a geometric sphere or an affine subspace of dimension $k$. A similar dichotomy holds for conformal expanding repellers.