Fino–Vezzoni conjecture for the pluriclosed flow when $c_1=0$

Let $(M,\omega_B)$ be a compact complex manifold with vanishing first Chern class, equipped with a balanced metric. Assume that $M$ admits a pluriclosed metric $\omega_0$, and let $\omega_t$ denote the pluriclosed flow with initial condition

$$\omega_{t\mid t=0}=\omega_0.$$

Fino–Vezzoni flow conjecture. The pluriclosed flow exists for all time and converges smoothly to a Kähler metric. In particular, $M$ is Kähler.

This conjecture is a proposed approach to the Fino–Vezzoni conjecture in the case $c_1(M)=0$. The source does not provide evidence resolving it; the paper verifies the asserted phenomenon only in a restricted invariant compact-quotient setting, so the general statement remains open.