Extremal Kähler cone non-closedness conjecture

Let the Kähler cone be the set of Kähler classes, and let the extremal Kähler cone be the subset consisting of classes represented by extremal Kähler metrics. Suppose the flow equation with initial data in the Kähler cone converges to a stationary point outside the Kähler cone. The extremal Kähler cone non-closedness conjecture. Then the extremal Kähler cone is not a closed subset of the Kähler cone. This relates failure of the flow to remain in the Kähler cone to the topology of the set of classes admitting extremal metrics.