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 K\"ahler metric as $t \to \infty$. 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.