Hausdorff measure conjecture for regular complex infinite IFS limit sets

Let $\mathcal{F}$ and $\mathcal{G}$ be regular conformal infinite iterated function systems (CIFS), with limit sets $J_{\mathcal{F}}$ and $J_{\mathcal{G}}$, Hausdorff dimensions $h_{\mathcal{F}}$ and $h_{\mathcal{G}}$, and parameters $\theta_{\mathcal{F}}$ and $\theta_{\mathcal{G}}$.

Hausdorff measure conjecture.

1. If $h_{\mathcal{F}}<2\theta_{\mathcal{F}}$, then

$$H_{h_{\mathcal{F}}}(J_{\mathcal{F}})=0.$$

2. If $h_{\mathcal{G}}<2\theta_{\mathcal{G}}$, then

$$H_{h_{\mathcal{G}}}(J_{\mathcal{G}})=0.$$

This would extend the preceding packing-measure results and is presented as a partial generalization of the cited theorem; its status is not established in the source.