Freeness conjecture for self-projective sets

Let $b[?m\mathcal{A}\subset\mathrm{SL}(2,\mathbb{R})b[0m$ be a finite set of matrices, let $b[?mG_{\mathcal{A}}b[0m$ be the semigroup it generates, and let $b[?m\Lambda_{\mathcal{A}}b[0m$ be its limit set. Let $b[?ms_{\mathcal{A}}b[0m$ denote the critical exponent associated with the zeta function of $b[?m\mathcal{A}b[0m$.

Freeness conjecture. If $b[?mG_{\mathcal{A}}b[0m$ is a free semigroup and $b[?m\Lambda_{\mathcal{A}}b[0m$ is non-empty and not a singleton, then

$$\dim_{\textup{H}}\Lambda_{\mathcal{A}}=\min\{1,s_{\mathcal{A}}\}.$$

This generalizes the exact-overlaps conjecture for self-similar sets to self-projective sets. It predicts that freeness is sufficient for the critical-exponent formula to give the Hausdorff dimension, beyond the special cases established earlier in the paper.