The two-tripod conjecture for smooth closed convex curves in spherical and hyperbolic geometry

Let $\gamma$ be a smooth closed convex curve in spherical or hyperbolic geometry. A tripod configuration is a configuration of three points on $\gamma$ satisfying the tripod condition studied in the paper.

Two-tripod conjecture. Every smooth closed convex curve in the spherical or hyperbolic geometry has at least two tripod configurations.

This would generalize the corresponding theorem for the planar, spherical, and hyperbolic settings and is presented as a natural extension of results in the planar case. The conjecture is stated without a resolution in the source.