Simon’s exact coincidence conjecture for self-similar iterated function systems

Let $\Phi = \{x\mapsto r_i x + a_i\}_{i\in {\mathcal I}}$ be a self-similar iterated function system on $\mathbb R$, with $|r_i|\in (0,1)$, and let $\Lambda_\Phi$ be its attractor. Write $\dim_{\rm \sim}(\Phi)$ for its similarity dimension. An exact overlap means that the semigroup generated by $\Phi$ is not free; equivalently, there exist $n\in {\mathbb N}$ and distinct words $u,v\in {\mathcal I}^n$ such that $\phi_u\equiv\phi_v$. Simon’s exact coincidence conjecture. If

$$\dim(\Lambda_\Phi)<\min\{1,\dim_{\rm \sim}(\Phi)\},$$

then $\Phi$ has an exact overlap. This conjecture predicts that dimension drop below the natural upper bound can occur only through exact algebraic coincidences among the iterated maps. Its status is not established by the supplied source context.