  An old problem asks whether bounded mild ancient solutions of the 3
dimensional Navier-Stokes equations are constants. While the full 3 dimensional
problem seems out of reach, in the works \cite{KNSS, SS09}, the authors
expressed their belief that the following conjecture should be true. For
incompressible axially-symmetric Navier-Stokes equations (ASNS) in three
dimensions: \textit{bounded mild ancient solutions are constant}.
  Understanding of such solutions could play useful roles in the study of
global regularity of solutions to the ASNS.
  In this article, we essentially prove this conjecture in the special case
that $u$ is periodic in $z$. To the best of our knowledge, this seems to be the
first result on this conjecture without unverified decay condition. It also
shows that periodic solutions are not models of possible singularity or high
velocity region. Some partial result in the non-periodic case is also given.
