Coincidence of the cotangent-bundle and diffeomorphism-group symplectic structures

Let $M=S^2$ be a two-dimensional sphere, let $\mathcal{D}_\mu$ be the group of area-preserving diffeomorphisms, and identify $T^*\mathcal{D}_\mu$ with $\mathcal{D}_\mu\times\Omega_0^2$ by right translations and the identification $\mathfrak{X}_\mu^*\simeq\Omega_0^2$. Under the corresponding identification, regard the diffeomorphism group $\mathcal{D}$ as a subset of $T^*\mathcal{D}_\mu$. Coincidence conjecture. The natural symplectic structure $W^{T^*}$ on the cotangent bundle $T^*\mathcal{D}_\mu$ coincides with the symplectic structure $W^\mathcal{D}$ on the diffeomorphism group $\mathcal{D}$, understood as a subset of $T^*\mathcal{D}_\mu$. This would identify the symplectic geometry arising from the diffeomorphism group with the canonical cotangent-bundle geometry in this Hamiltonian reduction; the supplied text does not indicate whether the assertion has been proved or disproved.