The Gauduchon canonical-connection Kähler-like conjecture

Let $M$ be a compact complex manifold endowed with a Hermitian metric, and let $abla^{\varepsilon}$ be a canonical connection in the Gauduchon family, distinct from the Strominger–Bismut connection and the Chern connection. The Gauduchon canonical-connection Kähler-like conjecture. If $abla^{\varepsilon}$ is Kähler-like, then the metric is Kähler. This conjecture extends the proposed characterization of Kähler metrics through Gauduchon-flat connections. It is motivated by the expectation that a generic canonical connection combines properties of the Chern and Strominger–Bismut connections, together with the conjectured incompatibility of balanced and pluriclosed metrics on non-Kähler compact complex manifolds; the general statement remains open.