Angles between simple, closed geodesics on convex surface

Question

It is known that there are at least three simple, closed geodesics on the surface of any smooth convex body $K$ in $\mathbb{R}^3$, the Lusternik-Schnirelmann Theorem (see links below for references). For $\gamma_1$ and $\gamma_2$ two such geodesics, let $\alpha_\max( \gamma_1, \gamma_2 )$ be the largest angle among all the crossings of $\gamma_1$ and $\gamma_2$. My question is:

Q. Is there a lower bound on $\alpha_\max$, over all simple, closed geodesics on $K$, over all $K$? Or is it possible that, for some $K$, all crossing angles of all geodesic pairs can be arbitrarily small?

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• “The famous Lusternik-Schnirelmann Theorem of the Three Closed Geodesics”
• Simple, closed geodesics in $\mathbb{S}^3$ manifold.
• Added: Angle between geodesics in hyperbolic surface.

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     ^{Image from the article Geodesics on an ellipsoid.}