Darboux theorem and symplectomorphisms

Question

In the lecture note I am reading there is following claim:

Let $(M,\omega)$ be a symplectic manifold, $f,g : M \rightarrow M$ symplectomorphisms, and $L \subset M$ a Lagrangian. Suppose $f(x) = g(x)$ for all $x \in L$. Then on a neighbourhood of $L$ there exists a symplectomorphim $h$ such that $ h f h^{-1} = g$.

Is this true? It says it follows from Darboux theorem but I don't believe it.

Answer 1

The claim is false. Consider two linear transformations of $R^2$: One is the identity, the other is a nontrivial shear along a straight line L.