  A cobordism between links in thickened surfaces consists of a surface $ S $
and a $3$-manifold $M $, with $ S $ properly embedded in $ M \times I $. We
show that there exist links in thickened surfaces such that if $(S,M) $ is a
cobordism between them in which $ S $ is simple, then $ M $ must be complex.
That is, there are cases in which low complexity of the surface does not imply
low complexity of the $3$-manifold.
  Specifically, we show that there exist concordant links in thickened surfaces
between which a concordance can only be realised by passing through thickenings
of higher genus surfaces. We exhibit an infinite family of such links that are
detected by an elementary method and other families of links that are not
detectable in this way. We investigate an augmented version of Khovanov
homology, and use it to detect these families. Such links provide
counterexamples to an analogue of the Slice-Ribbon conjecture.
