Maximal-time conjecture for the Kähler–Ricci flow coupled with a (1,1)-form

Let $(\mathcal{X},\omega)$ be a closed Kähler manifold, let $\alpha$ be a closed nonnegative $(1,1)$-form, and let $(\omega(t),\alpha(t))$ solve the coupled flow on its maximal interval $[0,T)$. Write $[\omega]$ and $[\alpha]$ for the corresponding cohomology classes, and interpret $A>0$, $A\leq B$, and $A\geq B$ for Kähler or nef classes as in the source. Maximal-time conjecture. The maximal time is

$$T=\sup\left\{t\in(0,+\infty]:(1-e^{-t})([\alpha]-[\omega]-2\pi c_1(\mathcal{X}))+[\omega]>0\right\}.$$

In particular, (i) if $[\omega]\leq[\alpha]-2\pi c_1(\mathcal{X})$, then $T=+\infty$; (ii) if $[\omega]>[\alpha]-2\pi c_1(\mathcal{X})$ and $[\alpha]\geq2\pi c_1(\mathcal{X})$, then $T=+\infty$; and (iii) if $[\omega]>[\alpha]-2\pi c_1(\mathcal{X})$ and $[\alpha]<2\pi c_1(\mathcal{X})$, then

$$T=\ln\left(1+\frac{[\omega]}{2\pi c_1(\mathcal{X})-[\alpha]}\right).$$

This conjecture predicts that the lifespan of the coupled flow is determined exactly by the evolving Kähler cohomology class, paralleling the corresponding characterization for Kähler–Ricci flow. The source does not report a resolution.