Scalar-curvature extension conjecture for the Kähler–Ricci flow coupled with a (1,1)-form

Let $(\mathcal{X},\omega)$ be a closed Kähler manifold and let $\alpha$ be a closed nonnegative $(1,1)$-form satisfying

$$[\omega]=-2\pi c_1(\mathcal{X})+[\alpha].$$

Let $(\omega(t),\alpha(t))$ solve the Kähler–Ricci flow coupled with $(1,1)$-forms on its maximal interval. Scalar-curvature extension conjecture. Under this topological condition, if $R_{\omega(t)}$ and $\operatorname{tr}_{\omega(t)}\alpha(t)$ are uniformly bounded, then the flow exists for all time. The conjecture is an analogue of the scalar-curvature extension question for Ricci flow. The source notes progress for the Calabi flow via an elliptic approach, while the parabolic approach remains open.