Angella–Otal–Ugarte–Villacampa conjecture for compact Bismut Kähler-like manifolds

Let $(M^n,g)$ be a compact Hermitian manifold. A Hermitian manifold is Bismut Kähler-like (BKL) if its Bismut curvature has the Kähler symmetries

$$R^b_{XYZ\,\overline{W}}=0, \qquad R^b_{X\,\overline{Y}Z\,\overline{W}}=R^b_{Z\,\overline{Y}X\,\overline{W}}$$

for all type $(1,0)$ tangent vectors $X,Y,Z,W$. AOUV conjecture. Every compact BKL manifold is pluriclosed; equivalently, BKL $\Longrightarrow$ pluriclosed. This conjecture concerns whether the curvature symmetries defining BKL force the pluriclosed condition, and the source presents it as the conjecture confirmed in a special setting by later work.