The Fino--Vezzoni conjecture on homogeneous spaces

We prove that a compact discrete quotient of a complex homogeneous space with compact isotropy is K"ahler whenever it admits both a balanced metric and a pluriclosed metric. Moreover, if its real first Chern class vanishes, then the pluriclosed flow starting from any invariant pluriclosed metric exists for all time and converges smoothly to a flat K"ahler metric.