Ergodicity conjecture for full-limit-set Kleinian groups

Let $\Gamma$ be a finitely generated Kleinian group, with limit set $\Lambda$ in the Riemann sphere. Ergodicity conjecture. If

$$\Lambda=\widehat{{\mathbf C}},$$

then $\Gamma$ acts ergodically on

$$\widehat{{\mathbf C}}\times\widehat{{\mathbf C}}.$$

This is described as a complementary conjecture associated with Ahlfors' Measure Conjecture, within the broader framework of rigidity and ergodicity for Kleinian groups. No resolution is stated in the provided text.