Shortest 2-design curve conjecture for the 2-sphere

Let $\gamma$ be the curve derived in the paper's Subsection~, considered up to isometry and reparameterization. A 2-design curve is a curve satisfying the spherical $2$-design condition.

Shortest 2-design curve conjecture. The curve $\gamma$ is the shortest $2$-design curve in $S^2$, up to isometry and reparameterization. The conjecture is motivated by the preceding construction; its validity and the classification of all equality cases remain open.