Conjecture on iterated balancing filtrations and Kähler–Ricci flow

Let $X$ be a K-semistable Fano manifold, and let $\omega$ denote the initial Kähler form for the Kähler–Ricci flow. The iterated balancing filtration is the filtration referred to in the paper as governing the asymptotics of the relevant geometric flows. Conjecture on iterated balancing filtrations. The asymptotics of the Kähler–Ricci flow starting from $\omega$ can be computed from the iterated balancing filtration. The Kähler–Ricci flow is not the gradient flow of the squared norm of a moment map, so the logarithmic asymptotic scale may need to be replaced by another function; the general assertion remains open.