Suppose I have a holomorphic symplectic manifold, and a smooth $(1,0)$ connection on the tangent bundle which is compatible with both the complex and the symplectic structures. Say that the associated curvature has type $(1,1)$. Then there is a Rozansky-Witten form associated to the Theta graph. It is a $\overline{\partial}$ closed $(0,2)$ form. Its associated cohomology class is a Rozansky-Witten class.

Can I replace the connection with another connection with the same properties such that the Rozansky-Witten form is actually $d$ closed? Is there some way to measure any obstructions to doing this?