The antipodal-pair conjecture for non-1-parameter line fibrations

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

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

A fibration is 1-parameter if it is a fibration of that type in the sense used in the paper. Antipodal-pair conjecture. If a fibration is not 1-parameter, there is at most one pair of antipodal points in $U$. If such a pair $\pm u$ exists, then $S_v$ is compact for every $v\in U-\{\pm u\}$, and $U-\{\pm u\}$ is convex.

The preceding discussion identifies exotic fibrations containing antipodal directions as the only known non-1-parameter examples with this feature. The conjecture asserts both uniqueness of the antipodal pair and convexity and compactness away from it; no resolution is supplied in the source.