Let $S$ be a closed hyperbolic surface of genus $g\geq 2$. Let $(\mathcal{T},\omega)$ be the corresponding Teichmuller space with the Weil–Petersson symplectic from $\omega$. Let $\Phi:\mathcal{T}\rightarrow\mathcal{T}$ be any diffeomorphism which preserves $\omega$.

Q) Does there exist a diffeomorphism $\phi:S\rightarrow S$ such that the induces map to $\mathcal{T}$ is $\Phi$?

Any suggestion or reference will be extremely helpful. Thanks in advance.