I will prove a rigidity result for Lagrangian cobordisms using an $L^2$ version of Floer homology. The input from holomorphic curves will be compressed in an easily understandable black box (at the expense of generality) and the construction of cellular $L^2$ homology will be described in some detail; therefore the talk should be accessible also to non symplectic geometers. This is a joint work with Baptiste Chantraine, Georgios Dimitroglu Rizell and Roman Golovko.