Scalar-flat K\"ahler surfaces whose Weyl tensor annihilates the Ricci form

We conjecture that any scalar-flat K\"ahler surface in which the Weyl tensor acting on 2-forms annihilates the Ricci form must be either Ricci-flat or locally isometric to a Riemannian product of two real surfaces with mutually opposite nonzero constant Gaussian curvatures. This amounts to the nonexistence of proper weakly Einstein anti-self-dual K\"ahler surfaces. We prove the above conjecture in three special cases: when the manifold is compact, when one of the Ricci eigendistributions is integrable, and when the norms of the Ricci and Weyl tensors are functionally dependent