A note on the pluriclosed flow on balanced manifolds with $c_1=0$

We conjecture that on any compact balanced manifold $(M,{\omega }_B)$ with $c_1(M)=0$, the pluriclosed flow admits long-time solutions ${\omega }_t$ for every initial pluriclosed metric, and that ${\omega }_t$ converges smoothly to a Kähler metric as $t\to \mathrm{\infty }$. We verify that this phenomenon occurs when $M$ is a compact quotient of a Lie group by a discrete subgroup, the background metric ${\omega }_B$ is invariant with vanishing Chern–Ricci form, and the initial metric ${\omega }_0$ is invariant. In particular, this provides new evidences for the Fino-Vezzoni conjecture.