The Monge–Ampère determination conjecture for Ricci-flat metrics on resolved threefolds

Let $Y_{[3]}^\Gamma$ be a resolved threefold with exceptional divisor $\mathcal{ED}$, and let the Ricci-flat metric and the Kähler metric on $\mathcal{ED}$ be those considered in the Kronheimer construction. Monge–Ampère determination conjecture. The Ricci-flat metric on $Y_{[3]}^\Gamma$ is completely determined in terms of a Monge–Ampère equation from the Kähler metric on the exceptional divisor $\mathcal{ED}$, as it is determined by the Kronheimer construction. The conjecture compares the Ricci-flat metric with the Kronheimer construction and asserts that the metric is recoverable from the induced Kähler geometry of the exceptional divisor; the supplied passage does not state what cases are known or whether the conjecture has been resolved.