Desingularizations of Conformally Kaehler, Einstein Orbifolds

Suppose we are given a sequence of smooth compact oriented Einstein 4-manifolds with fixed positive Einstein constant that Gromov-Hausdorff converges to a 4-dimensional Einstein orbifold. Suppose, moreover, that the limit metric is Hermitian with respect to some integrable complex structure on the limit orbifold, and that this orbifold has at least one singular point. Finally, suppose that every gravitational instanton that bubbles off from the sequence is anti-self-dual. Then, far out in the sequence, all the given Einstein manifolds are actually Kaehler-Einstein. As a consequence, the limit orbifold is also Kaehler-Einstein, and must in fact be one of the orbifold limits classified by Odaka, Spotti, and Sun.