Modified Calabi-flow convergence conjecture for extremal metrics

Let $X$ be a Kähler manifold with Kähler class $[\omega]$. Suppose that there exists an extremal metric $\omega_0\in[\omega]$. Let $\omega_1\in[\omega]$ be a Kähler metric invariant under the maximal compact subgroup of the identity component of the reduced automorphism group. Assume that the curvature along the Calabi flow starting from $\omega_1$ is uniformly bounded. Modified Calabi-flow convergence conjecture. The modified Calabi flow converges to an extremal metric exponentially fast. This conjecture concerns convergence after accounting for automorphisms when the class contains an extremal metric. The paper presents it as a conjecture and later states that its toric analogue is proved by the main theorem; no general resolution is supplied here.