Ricci-vertex proximity conjecture for Kähler-Ricci flow fibres

Let $g(t)$ be the maximal Kähler-Ricci flow on $X\times[0,1)$, let $\Phi:X\to Y$ be the limiting fibration, and let $\mathcal{F}_q=\Phi^{-1}(q)$. A Ricci vertex associated to a smooth closed $(1,1)$-form $\theta_Y\in\vartheta$ at time $t$ is denoted by $p_t$. Ricci-vertex proximity conjecture. For every $q\in Y$, there exist such a form $\theta_Y$ and $C=C(n,g_0,\theta_Y)>0$ such that, for all $t\in[0,1)$,

$$d_{g(t)}(p_t,\mathcal{F}_q)\leq C(1-t)^{1/2}.$$

This would control the location of Ricci vertices relative to every fibre and is confirmed in the paper only in the Kähler-Ricci flows with large symmetry.