The virtual circle-action conjecture for lattices in SO(1,n)

Let $\Gamma$ be any lattice in $\mathop{\mathrm{SO}}\nolimits(1,n)$. Virtual circle-action conjecture. Some finite-index subgroup of $\Gamma$ has a $C^\infty$ action on $S^1$ with no finite orbits. The conjecture follows from Thurston's stated virtual first-Betti-number conjecture because $\mathbb{Z}$ has a $C^\infty$ circle action with no finite orbits; it is open.