Zhao–Zheng's two canonical connections conjecture for different Hermitian metrics

Let $(M^n,g,g')$ be a compact Hermitian manifold with possibly different Hermitian metrics $g$ and $g'$. Let $D^r_s$ be the canonical metric connection associated with $g$, and let $D^{r'}_{s'}$ be the corresponding connection associated with $g'$, where $(r,s)$ and $(r',s')$ are distinct points of $\Omega$. Exclude the four exceptional pairs listed in Theorem 5 of the source. A connection is Kähler-like when its curvature has the Kähler curvature symmetries. Zhao–Zheng's conjecture. If both $D^r_s$ and $D^{r'}_{s'}$ are Kähler-like, then $M^n$ admits a Kähler metric. The conjecture extends the preceding result for two canonical connections on the same metric to possibly different metrics; the source does not state a resolution.