Constant linear-combination conjecture for the symplectic forms on the moduli space of connections

Let ${\mathcal H}_g(r)$ be the moduli space of the vector-bundle and connection data considered in the paper, equipped with the holomorphic symplectic form from Corollary $(1)$. The projection to the moduli space of projective structures gives two closed holomorphic $2$-forms on ${\mathcal H}_g(r)$: one induced by the natural holomorphic symplectic form on ${\mathcal D}_X(r)$, and one induced by the holomorphic symplectic form on ${\mathcal P}_g$. Constant linear-combination conjecture. The holomorphic symplectic form on ${\mathcal H}_g(r)$ in Corollary $(1)$ is a constant linear combination of the above two holomorphic $2$-forms on ${\mathcal H}_g(r)$. The conjecture concerns the precise relationship among the symplectic structures arising from connections and projective structures; the supplied text does not indicate whether it has been proved or disproved.