Uniform scalar-curvature bounds along the Kähler–Ricci flow

Prove that, for a compact normal Kähler variety with klt singularities and an initial Kähler current whose Ricci current is bounded above, the scalar curvature of the associated weak Kähler–Ricci flow remains uniformly bounded on $X\times[0,T-\epsilon]$ for every $\epsilon\in(0,T)$.

Background

In addition to the conjectured metric convergence and homeomorphism statement for weak Kähler–Ricci flow on klt varieties, the authors formulate a further conjecture under the stronger assumption that the initial Ricci current is bounded above.

The unresolved claim concerns a uniform scalar-curvature estimate away from the terminal time. It is presented as an additional conjectural regularity property of the flow and is not proved in the paper.