Atiyah's faithfulness conjecture for the hyperbolic-space action on the sphere-at-infinity measure

Let $\omega$ be the symmetric measure on the sphere at infinity $S^2$ of hyperbolic space ${\mathbb H}^3$. For $f\in C^0_c({\mathbb H}^3)$, define the action

$$f\cdot\omega=\int_{{\mathbb H}^3}f(x)\,x\cdot\omega\,dx,$$

where $x$ is regarded as an element of $SL(2,{\mathbb C})/SU(2)$. Atiyah's faithfulness conjecture. For every $f\in C^0_c({\mathbb H}^3)$,

$$f\cdot\omega=0\quad\Longleftrightarrow\quad f\equiv 0.$$

The conjecture asserts that this action is faithful: the boundary measure detects every compactly supported continuous function on hyperbolic space. The source attributes the suggestion to Atiyah and gives no resolution.