Does a compact nonflat surface without conjugate points have ergodic geodesic flow?

PDF download of cited [BBB87] J. Diff. Geom. 1987 paper.

I cannot resist posting their $6$-legged dinosaur(?) Fig.2:

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"[W]e arrange that the geodesic $\gamma_0$ passing through the centers of the caps is positively and negatively asymptotic to closed geodesics $\sigma_{+}$ and $\sigma_{-}$ which do not meet the caps (see Figure 2).

This is an open question. Actually it is still not known if a compact non-flat surface with non-positive curvature has an ergodic geodesic flow (with respect to the Liouville measure).