Geometrization conjecture for pluriclosed flow on properly elliptic surfaces

Let $(M^4,J)$ be a properly elliptic surface with $\operatorname{kod}(M)=1$ and odd first Betti number, and let $\omega$ be a pluriclosed metric. Properly elliptic geometrization conjecture. The solution to pluriclosed flow with this initial data exists on $[0,\infty)$; $(M,\omega_t/t)$ converges in the Gromov–Hausdorff topology to $(B,\omega_{KE})$, the base curve with its canonical orbifold Kähler–Einstein metric; and on the universal cover there is a blowdown limit $\widetilde{\omega}_{\infty}(t)=\lim_{s\to\infty}s^{-1}\widetilde{\omega}(st)$ that is a locally homogeneous expanding soliton. The passage notes that the torus-invariant case is proved, while the general claim is conjectural.