= Solution
Take the usual convention that the <smooth isotopy> starts at $\rho_0=\mathrm{id}$. The <exterior derivative> commutes with a <pullback of a differential form>, so
$$
d(\rho_t^*\lambda-\lambda)
=\rho_t^*d\lambda-d\lambda
=\omega-\rho_t^*\omega.
$$
Consequently
$$
\boxed{\rho_t^*\lambda-\lambda\text{ is closed for every }t
\iff \rho_t^*\omega=\omega\text{ for every }t.}
$$
The right-hand condition says precisely that $\rho_t$ is a <symplectic isotopy>.
Here an <exact symplectic manifold> is necessarily noncompact in positive dimension: if $\dim M=2n$ and $\omega=-d\lambda$, then $\omega^n=-d(\lambda\wedge\omega^{n-1})$, contradicting the positive <symplectic volume> and the <Generalized Stokes theorem> on a <compact> <smooth manifold> without boundary. Thus the paper's introductory word “closed” must here be read as “without boundary”, as its parenthesis suggests, rather than as including <compactness>.
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