The parametrised Whitehead torsion is an invariant of families of manifolds, and can be viewed as a map to an algebraic K-theory space. A strong version of the nearby Lagrangian conjecture says that when applied to families of closed exact Lagrangians in a cotangent bundle, this invariant vanishes.Abouzaid and Kragh showed that in this case, this map lands in the trivial path component of the target, i.e. is trivial on π0. Using generating functions, we find strong constraints on what this map does to higher homotopy groups. I'll illustrate this with some concrete consequences for the symplectic mapping class group of T∗Tnrelative to the 0-section.This is based on joint work-in-progress with Sylvain Courte.