Conjecture on the uniqueness of the type of Bach-flat Kähler metrics

Let $(M,J)$ be a compact complex surface. A Bach-flat Kähler metric is a Kähler metric compatible with $J$ whose Bach tensor vanishes. The type of such a metric is one of the types in the classification scheme of Theorem. Same-type conjecture. On a fixed compact complex surface $(M,J)$, any pair of $J$-compatible Bach-flat Kähler metrics are necessarily of the same type, in the sense of Theorem. The preceding proposition shows that metrics of different types, if they exist, must be of types II(b) and III(b); the conjecture asserts that this remaining possibility does not occur.