Absolute continuity conjecture for contracting-on-average self-similar measures

Let $G$ be the group acting on $\mathbb{R}^d$ in the self-similar setting, and let $\nu$ be the associated stationary measure of a finitely supported, contracting on average and irreducible probability measure $\mu$ on $G$ without a common fixed point. Write $h_{\mu}$ for the entropy of $\mu$ and $\chi_{\mu}$ for its Lyapunov exponent. Absolute continuity conjecture. Then $\nu$ is absolutely continuous if

$$\frac{h_{\mu}}{|\chi_{\mu}|}>d.$$

This conjecture predicts absolute continuity in the dimension-saturated regime beyond the natural entropy-to-Lyapunov threshold. The preceding dimension results establish the expected dimension formula under additional separation or arithmetic hypotheses, but the stated absolute-continuity implication is presented as folklore and remains open in this generality.