Tosatti–Weinkove's Chern-Ricci flow convergence conjecture for minimal surfaces

Let $X^n$ be a compact complex manifold with $K_X$ nef and big, and let $tilde\omega(s)$ be the solution of the Chern-Ricci flow

$$\frac{\partial}{\partial s}\widetilde\omega(s)=-\operatorname{Ric}^C(\widetilde\omega(s)),\qquad \widetilde\omega(0)=\omega_0,$$

starting at an arbitrary Hermitian metric $\omega_0$. Let $E=\operatorname{Null}(K_X)$, and let $\omega_{KE}$ denote the singular Kähler–Einstein current obtained as the limit of the normalized Chern-Ricci flow. Tosatti–Weinkove's conjecture. There is a constant $C$ such that

$$\operatorname{diam}\left(X,\frac{\widetilde\omega(s)}{s+1}\right)\leq C$$

for all sufficiently large $s$, and

$$\left(X,\frac{\widetilde\omega(s)}{s+1}\right)\longrightarrow (Z,d)$$

in the Gromov–Hausdorff topology for some compact metric space $(Z,d)$. Moreover, $(Z,d)$ should be identified with the metric completion of $(X\setminus E,\omega_{KE})$. On surfaces of general type, this conjecture was proved independently by Guo–Song–Weinkove and Tian–Zhang; Tian–Zhang also treated the Kähler case in complex dimension three.