I will report on a joint work with M. Abouzaid, S. Guillermou and T. Kragh. The nearby Lagrangian conjecture predicts that a closed exact Lagrangian submanifold in a cotangent bundle must be Hamiltonian isotopic to the zero-section. In particular the stable Gauss map of such a Lagrangian should be null-homotopic. I will explain a proof of the weaker statement that the stable Gauss map vanishes on all homotopy groups. The key concept that we introduce is the notion of twisted generating function designed to describe Lagrangians with a priori non trivial Gauss map.We also rely on deep results of pseudo-isotopy theory due to M. Bökstedt.