Podestà–Zuddas conjecture on invariant Bismut-parallel metrics on complex flag manifolds

Let $(G/H,J)$ be a complex flag manifold with $J$ a fixed $G$-invariant complex structure, and let a $G$-invariant BTP metric mean a $G$-invariant Hermitian metric whose Bismut torsion is parallel. Let $g_o$ denote the standard metric.

Podestà–Zuddas conjecture. Every $G$-invariant BTP metric on $(G/H,J)$ is either a Kähler metric or a multiple of the standard metric $g_o$.

The conjecture generalizes the known classification for flag manifolds with at most two isotropy summands and for complex full flag manifolds of the form $SU(n+1)/T^n$. In the supplied paper it is introduced as a conjecture to be proved, so its status in the source context is open.