Tameness conjecture for the rolling-surface $G_2/P_2$-geometry

Let $S$ and $S'$ be Riemannian surfaces whose Gauss curvatures differ at every pair of points, and let $M$ be the space of linear isometries between tangent planes of $S$ and $S'$. The nondegenerate $2$-plane field on $M$ carries a $G_2/P_2$-geometry. Let $L$ and $L'$ denote the common lengths of the closed geodesics on $S$ and $S'$, respectively, when those common lengths exist. Rolling-surface tameness conjecture. The $G_2/P_2$-geometry on $M$ is tame if and only if both surfaces are Zoll surfaces, meaning that all their geodesics are closed and embedded curves, and $L$ is a rational multiple of $L'$. The conjecture proposes a precise relation between tameness and the global geodesic dynamics of the two surfaces; the supplied text gives no resolution.