Tosatti's conjecture on independence of Kähler–Ricci flow singularity type

Let $X$ be a compact Kähler manifold with nef canonical bundle $K_X$, so every solution of the Kähler–Ricci flow exists for all positive time. For a long-time solution, a singularity type at infinity is type III if

$$\sup_{X\times[0,\infty)}|\operatorname{Rm}(\omega(t))|_{\omega(t)}<+\infty,$$

and type IIb if

$$\sup_{X\times[0,\infty)}|\operatorname{Rm}(\omega(t))|_{\omega(t)}=+\infty.$$

Tosatti's conjecture. The singularity type at infinity does not depend on the choice of the initial metric $\omega_0$.

The conjecture asserts that the long-time curvature behavior is determined by the underlying manifold rather than by the initial Kähler metric. The supplied source does not indicate that this independence has been proved or disproved.