Geometrization conjecture for pluriclosed flow on Inoue surfaces

Let $(M^4,J)$ be an Inoue surface, and let $\omega_0$ be a pluriclosed metric on $M$. Inoue surface geometrization conjecture. The solution $\omega_t$ to pluriclosed flow with this initial data exists on $[0,\infty)$; $(M,\widehat{\omega}_t/t)$ converges as $t\to\infty$ to a circle in the Gromov–Hausdorff topology whose length depends only on the complex structure; 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 canonical locally homogeneous expanding soliton. The passage describes homogeneous and commuting generalized Kähler cases as known, but leaves the general behavior conjectural.