Fibre diameter and scalar curvature conjecture near Ricci vertices

Let $g(t)$ be the maximal Kähler-Ricci flow on $X\times[0,1)$, let $\theta_Y\in\vartheta$ be a smooth closed $(1,1)$-form on $Y$, and let $p_t$ be a Ricci vertex associated to $\theta_Y$ at time $t$. For $D>0$, let $B(p_t,t,D(1-t)^{1/2})$ denote the time-$t$ metric ball of that radius. For $q$ in this ball, set $\mathcal{F}_q=\Phi^{-1}(\Phi(q))$. Fibre diameter and scalar curvature conjecture. There is $C=C(n,g_0,\theta_Y,D)>0$ such that

$$\operatorname{Diam}(\mathcal{F}_q,g(t))\leq C(1-t)^{1/2},$$

and

$$\sup_{\mathcal{F}_q}|R(\cdot,t)|\leq \frac{C}{1-t}.$$

Here $R(t)$ is the scalar curvature and the diameter is measured in $(X,g(t))$. This conjecture seeks to remove the tubular-neighbourhood volume assumption used in the preceding theorem; its general validity is open.