Are lines much bigger than line segments?

We pose the following conjecture: () If A is the union of line segments in Rⁿ, and B is the union of the corresponding full lines, then the Hausdorff dimensions of A and B agree. We prove that this conjecture would imply that every Besicovitch set (compact set that contains line segments in every direction) in Rⁿ has Hausdorff dimension at least $n-1$ and (upper) Minkowski dimension n. We also prove that conjecture () holds if the Hausdorff dimension of B is at most $2$, so in particular it holds in the plane.