  We study Lagrangian cobordism groups of closed symplectic surfaces of genus
$g \geq 2$ whose relations are given by unobstructed, immersed Lagrangian
cobordisms. Building upon work of Abouzaid and Perrier, we compute these
cobordism groups and show that they are isomorphic to the Grothendieck group of
the derived Fukaya category of the surface.
