Jian–Song metric identification conjecture for Kähler–Ricci flow limits

Let $X$ be an $n$-dimensional Kähler manifold with semi-ample canonical line bundle $K_X$, and let $g(t)$ be the long-time solution of the normalized Kähler–Ricci flow. For a sequence $t_j\to\infty$, let $(X,g(t_j))$ converge in Gromov–Hausdorff topology to $(X_\infty,d_\infty)$. Let $(Y,d_Y)$ be the metric completion of the regular canonical model $(X_{\textnormal{can}}^\circ,g_{\textnormal{can}})$, and let $\Psi:Y\to X_\infty$ and $\Upsilon:X_\infty\to X_{\textnormal{can}}$ be the Lipschitz extensions of the identity maps described in the source. Jian–Song metric identification conjecture. The map $\Psi$ is an isometry and the map $\Upsilon$ is a homeomorphism. This conjecture concerns the precise metric and topological identification of Gromov–Hausdorff limits with the canonical model. The stated homeomorphism results are known when $\operatorname{kod}(X)=0,1,2,n$, while the general case remains open.