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 (the graph that looks like the letter theta).  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?