Conjecture on lower-order controls and generalized Kähler–Einstein limits for Kähler–Ricci flow

Consider the Kähler–Ricci flow with infinite-time singularities, meaning that $[L]$ is nef but not Kähler, on a compact Kähler manifold $X$ with initial Kähler form $5bomega_0$. Let $u$ be the scalar potential of the flow, and let $PSH_L(X)$ denote the set of $L$-plurisubharmonic functions. Suppose $k\in\{0,\ldots,n\}$ is the smallest integer such that

$$[L]^{n-k}\cdot[\omega_0]^k\neq 0.$$

Lower-control and limit conjecture. Uniformly on $X\times[0,\infty)$, there is a constant $C$ such that

$$\frac{\partial u}{\partial t}\geq -C.$$

Moreover, for every $\varphi\in PSH_L(X)$, there is a constant $C$ such that

$$u\geq -kt-C+\varphi.$$

Furthermore, $u+kt$ converges in some proper sense to a function $\Phi\in PSH_L(X)$ with minimal singularities, and $L+\sqrt{-1}\partial\bar\partial\Phi$ is the generalized Kähler–Einstein current associated with $[L]$. This conjecture seeks a differential-geometric approach to the Abundance Conjecture by describing potential growth and the limiting current in the infinite-time singularity case; the asserted uniform estimates and convergence remain unresolved in the source.