  In the paper \cite{KNSS:1}, the authors make the following conjecture: {\it
any bounded ancient mild solution of the 3D axially symmetric Navier-Stokes
equations is constant.} And it is proved in the case that the solution is swirl
free. Our purpose of this paper is to improve their result by allowing that the
solution can grow with a power smaller than 1 with respect to the distance to
the origin. Also, we will show that such a power is optimal to prove the
Liouville type theorem since we can find counterexamples for the Navier-Stokes
equations such that the Liouville theorem fails if the solution can grow
linearly.
