The great-circle conjecture for noncompact base directions

Let $V:\mathbb R^3\to S^2$ be a unit vector field defining a fibration of $\mathbb R^3$ by oriented lines. Put $U=V(\mathbb R^3)$, and for $u\in U$ let $u^\perp$ be the plane through the origin orthogonal to $u$ and define

$$S_u=u^\perp\cap V^{-1}(u).$$

Great-circle conjecture. If there exist multiple $u\in U$ for which $S_u$ is noncompact, then the set of such $u$ is contained in some great circle in $S^2$.

The theorem preceding this conjecture shows that the noncompact base spaces are precisely the source of the less-understood behavior not covered by the all-compact or all-noncompact cases. The conjecture concerns the possible directions of these noncompact base spaces; its resolution is not indicated here.