The type I singularity conjecture for Kähler-Ricci flow on Kähler surfaces

Let $(M,g(t))$ be a Kähler-Ricci flow on a Kähler surface, defined up to a maximal existence time $0<T\leq\infty$. A finite-time singularity occurs when $T<\infty$; it is type I if

$$\limsup_{t\to T}(T-t)\sup_{x\in M}|\operatorname{Rm}(x,t)|<\infty.$$

Type I singularity conjecture. Every finite-time singularity of the Kähler-Ricci flow on a Kähler surface is type I.

Finite-time singularities are classified by the blow-up rate of the Riemann curvature, with type II singularities characterized by the corresponding limsup being infinite. The conjecture is motivated by results for several classes of Kähler surfaces, including Hirzebruch surfaces and blow-ups, but the general assertion remains open.