Convergence conjecture for the modified Calabi flow on relatively K-stable toric varieties

Let $(X,L)$ be a polarized toric variety, with $L$ an ample line bundle, and let $c_1(L)$ be its first Chern class. Assume that $(X,L)$ is relative $K$-stable, meaning relatively K-stable in the sense used for polarized varieties. Let $\omega\in c_1(L)$ be a Kähler metric invariant under the toric action. Assume that the Calabi flow starting from $\omega$ exists for all time and that the Riemannian curvature is uniformly bounded along the flow.

Toric modified-Calabi-flow conjecture. The modified Calabi flow converges exponentially fast to an extremal metric in $c_1(L)$.

This conjecture is the toric, symmetry-restricted convergence statement linking relative K-stability to extremal metrics. The source gives no general resolution.