Finite-time scalar curvature and fibre diameter conjecture for Kähler-Ricci flow

Let $g(t)$ be the smooth solution of the Kähler-Ricci flow on $X\times[0,T)$, where $T<\infty$, and let $\Phi:X\to Y$ be the map induced by the limiting semi-ample class. For $q\in Y$, write $\Phi^{-1}(q)$ for its fibre. Finite-time scalar curvature and fibre diameter conjecture. There exists $C<\infty$ such that for all $(x,t)\in X\times[0,T)$,

$$|R(x,t)|\leq \frac{C}{T-t},$$

and

$$\operatorname{Diam}(\Phi^{-1}(q),g(t))\leq C(T-t)^{1/2}.$$

Here $R(t)$ is the scalar curvature and the diameter is computed in $(X,g(t))$. This is proposed as an extension of Perelman’s finite-time estimates from the Fano case to general finite-time Kähler-Ricci flows; its general validity is open.