Connectedness conjecture for Anosov metrics on negatively curved 3-manifolds

Let $(\Sigma,g_0)$ be a negatively curved oriented closed $3$-manifold, and let the space of Anosov metrics on $\Sigma$ consist of metrics for which the geodesic flow is Anosov.

Connectedness conjecture. The space of Anosov metrics on $\Sigma$ is connected.

Connectedness would extend the generic conclusions established on the component containing a hyperbolic metric to all Anosov metrics on the manifold. The source says that the conjecture is open and notes a relationship with the conjectured long-time convergence of cross curvature flow.