  For an embedded exact Lagrangian cobordism between Legendrian submanifolds in
the 1-jet bundle, we prove that a generating family linear at infinity on the
Legendrian at the negative end extends to a generating family linear at
infinity on the Lagrangian cobordism after stabilization if and only if the
formal obstructions vanish. In particular, a Lagrangian filling with trivial
stable Lagrangian Gauss map admits a generating family linear at infinity.
