Canonical limit current conjecture for the Kähler-Ricci flow

Let $(X^n,\omega_0)$ be a compact Kähler manifold with $K_X$ nef and numerical dimension $0<m<n$, and let $\omega(t)$ be the immortal solution of the normalized Kähler-Ricci flow. Let $\chi$ represent $c_1(K_X)$.

Canonical limit current conjecture. There is a quasi-plurisubharmonic function $\varphi_\infty$ such that

$$\chi_\infty:=\chi+i\partial\bar\partial\varphi_\infty\geqslant0$$

weakly, $\varphi(t)\to\varphi_\infty$ in $L^1(X)$ as $t\to+\infty$, and consequently $\omega(t)\to\chi_\infty$ weakly as currents. Moreover, $\chi_\infty$ is independent of the initial metric $\omega_0$.

The claim asserts uniqueness and initial-metric independence of the subsequential limiting potential/current. The preceding theorem supplies subsequential limits with minimal singularities, while the full convergence and independence asserted here remain open in the stated generality; the semiample case has stronger known regularity.