The Complete Overlap Conjecture for self-similar attractors

Let $\mathcal{S}=\left\{S_i\right\}_{i=1}^{m}$ be a self-similar iterated function system on $\mathbb{R}^d$, with attractor $\Lambda$, similarity dimension $s$, and symbolic space $\Sigma^*$ of finite words. For a word $\mathbf{i}$, let $S_{\mathbf{i}}$ denote the corresponding composition of maps, and write $S_{\mathbf{i}}\equiv S_{\mathbf{j}}$ when the two compositions are identical. Complete Overlap Conjecture. The Hausdorff dimension satisfies

$$\dim_{\rm H}(\Lambda)<\min\left\{d,s\right\}$$

if and only if there exist distinct words $\mathbf{i},\mathbf{j}\in\Sigma^*$ such that $S_{\mathbf{i}}\equiv S_{\mathbf{j}}$. In the stated candidate, the IFS is on $\mathbb{R}$, so $d=1$.

This conjecture asserts that dimension drop below the natural upper bound occurs exactly because of an exact overlap between two cylinder maps. The source states that the conjecture does not hold in $\mathbb{R}^2$, so the claim is refuted as stated.