Biran and Cornea showed that Lagrangian cobordisms satisfying suitable geometric constraints give rise to algebraic decompositions in the derived Fukaya category DFuk(M) of a symplectic manifold M. They asked whether any such decomposition in DFuk(M) comes from a Lagrangian cobordism in this way. The goal of this talk is to answer this question in the case where M is a symplectic surface of higher genus. We consider a cobordism group whose relations are given by unobstructed, immersed Lagrangian cobordisms. We will sketch the proof that this unobstructed cobordism group is isomorphic to the Grothendieck group of DFuk(M), which gives a positive answer to Biran and Cornea's question at the level of K-theory. The proof builds upon work of Perrier, as well as the computation by Abouzaid of the Grothendieck group of DFuk(M).