The small-or-perfect geodesic minimizer conjecture for G-alpha

Let $G_\alpha$, with $-1\leq\alpha\leq 1$, be the homogeneous Riemannian 3-manifold equipped with its canonical left-invariant metric. For a geodesic segment $\gamma$ of length $T$ in $G_\alpha$, let $P_\gamma$ denote the period associated with its initial tangent vector, and call $\gamma$ small if $T<P_\gamma$ and perfect if $T=P_\gamma$. The small-or-perfect geodesic minimizer conjecture. A geodesic segment in $G_\alpha$ is a length minimizer if and only if it is small or perfect. This is presented as the primary aim in the study of the interpolation between Sol, $\mathbb{H}^2\times\mathbb{R}$, and $\mathbb{H}^3$; the supplied text gives no resolution of the conjecture.