I read this as a conjecture in the paper by Ballmann-Brin-Burns, titled "On Surfaces with No Conjugate points" JDG 25(249-273), 1987.


What is current status of this conjecture? 



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


