Brin's conjecture on ergodicity of the frame flow

Let $(M,g)$ be a negatively curved Riemannian manifold whose sectional curvature satisfies

$$-C \leq \kappa_g(u \wedge v) \leq -C\delta$$

for some constant $C>0$ and some $\delta\in(0,1]$; such a metric is called $\delta$-pinched. The frame flow is the natural flow on the frame bundle $FM$ induced by the geodesic flow and parallel transport.

Brin's conjecture. If $(M,g)$ is $\delta$-pinched for some $\delta>0.25$, then the frame flow is ergodic.

Brin proved ergodicity in dimension $3$, and Brin–Gromov proved it for odd dimensions other than $7$. The conjecture addresses the remaining even-dimensional cases, where Kähler manifolds provide non-ergodic examples at pinching at most $0.25$.