AOUV's two Gauduchon connections conjecture

Let $(M^n,g)$ be a compact Hermitian manifold, and let $D^r$ and $D^{r'}$ be Gauduchon connections for real parameters $r$ and $r'$. A connection is Kähler-like when its curvature has the Kähler curvature symmetries. AOUV's conjecture. If $r\neq r'$ and both $D^r$ and $D^{r'}$ are Kähler-like, then $g$ must be Kähler. This asserts that Kähler-likeness for two distinct members of the Gauduchon family forces the underlying metric to be Kähler; the source does not state a resolution.