On the primitivity of birational transformations of irreducible holomorphic symplectic manifolds

Irreducible holomorphic symplectic (IHS) manifolds are a higher dimensional analogue of K3 surfaces; if $X$ is such a manifold, we can define a quadratic form on $H^2(X,\mathbb Z)$ that bears a formal resemblance to the intersection product on a surface. 

A birational transformation $f$ of a manifold $X$ is said "imprimitive" if it preserves the fibres of a non-trivial fibration $\pi\colon X\dashrightarrow B$. Analogously to the surface case (Gizatullin), I will show that, if $X$ is IHS and $f$ induces a linear automorphism of $H^2(X,\mathbb Z)$ with at least an eigenvalue with modulus different then $1$, then it is primitive.