Left-handed geodesic flow of spheres of revolution

A flow on a 3-manifold is left-handed if any two ergodic invariant measures have negative linking number. We prove that on a 2-sphere of revolution whose curvature is $1\mathrm{/}4$-pinched the geodesic flow is left-handed. Conversely, for every $\delta \le 1\mathrm{/}4$, we construct a 2-sphere whose curvature is $\delta$-pinched and whose geodesic flow is not left-handed. This gives credit to a conjecture of Florio and Hryniewicz that $1\mathrm{/}4$ is the optimal pinching constant for left-handedness among arbitrary positively curved 2-spheres.