Must a closed manifold admitting a metric without conjugate points admit a metric of nonpositive curvature?

Theorem 1.1: there is a closed connected orientable smooth 3-manifold with a smooth metric without conjugate points that admits no smooth metric of nonpositive sectional curvature. The manifold glues two copies of (punctured torus) x $S^1$ by Leeb's obstructed gluing; $\pi_1$ admits no proper cocompact action on a CAT(0) space, so there is no locally CAT(0) metric, and by Ivanov-Kapovitch no metric without focal points. Not shown: higher-dimensional examples, or an answer to Ivanov-Kapovitch's other questions about fundamental groups.

Theorem 1.1: there is a closed connected orientable smooth 3-manifold with a smooth metric without conjugate points that admits no smooth metric of nonpositive sectional curvature. The manifold glues two copies of (punctured torus) x $S^1$ by Leeb's obstructed gluing; $\pi_1$ admits no proper cocompact action on a CAT(0) space, so there is no locally CAT(0) metric, and by Ivanov-Kapovitch no metric without focal points. Not shown: higher-dimensional examples, or an answer to Ivanov-Kapovitch's other questions about fundamental groups.