Let $X,Y$ be symplectic manifolds of dimensions at least 4 (in dimension 2 this is clearly false as pointed out by @Arctic Char in the comments) with symplectic forms $\omega_X,\omega_Y$ and let $f\colon X\rightarrow Y$ be a symplectic diffeomorphism, $f^\ast \omega_Y=\omega_Y$. It is a quite trivial fact that $f(\Lambda)$ is a Lagrangian submanifold in $Y$ whenever $\Lambda$ is Lagrangian in $X$, and similarly if $\Lambda$ is Lagrangian in $Y$ then $f^{-1}(\Lambda)$ is Lagrangian in $X$.
Question: Is the converse true? Namely, if $f$ is a diffeomorphism preserving Lagrangian submanifolds in both directions, if $f$ necessarily symplectic?
I know this is false is linearity is assumed throughout (namely, $X,Y$ symplectic vector spaces, $f$ linear and $\Lambda$ a vector subspace). What about asking that every submanifold is preserved? This would at least render the standard counterexample of $f(v)=2v$ inapplicable since there are clearly Lagrangian submanifolds of $\mathbb{R}^n$ which are not preserved. Also, this is false if only "conic Lagrangians" (homogeneous in the fibre directions) in $T^\ast X\setminus \{0\}$ and homogeneous symplectomorphisms are considered (again, just take dilations in the fibres).
However, I could find no statement nor explicit counterexample about maps preserving every Lagrangian submanifold. Furthermore, in the contact world the analogous statement is true, namely: If $X, Y$ are contact manifolds and $f\colon X\rightarrow Y$ is a diffeomorphism which preserves all Legendrian submanifolds, then $f$ is a contact diffeomorphism and vice-versa. This is proven in the 1964 paper by Sasaki, "A characterisation of contact transformations". Given the close relation and many analogies between contact and symplectic geometry, one at least hope that this be true.
Thanks in advance for any insight/reference/answer.