  In this paper, we consider the uniqueness of solutions to the 3d
Navier-Stokes equations with initial vorticity given by $\omega_0 = \alpha e_z
\delta_{x = y = 0}$, where $\delta_{x=y= 0}$ is the one dimensional Hausdorff
measure of an infinite, vertical line and $\alpha \in \mathbb R$ is an
arbitrary circulation. This initial data corresponds to an idealized, infinite
vortex filament. One smooth, mild solution is given by the self-similar Oseen
vortex column, which coincides with the heat evolution. Previous work by
Germain, Harrop-Griffiths, and the first author implies that this solution is
unique within a class of mild solutions that converge to the Oseen vortex in
suitable self-similar weighted spaces. In this paper, the uniqueness class of
the Oseen vortex is expanded to include any solution that converges to the
initial data in a sufficiently strong sense. This gives further evidence in
support of the expectation that the Oseen vortex is the only possible mild
solution that is identifiable as a vortex filament. The proof is a 3d variation
of a 2d compactness/rigidity argument in $t \searrow 0$ originally due to
Gallagher and Gallay.
