On Gauduchon connections with K\"ahler-like curvature

We study Hermitian metrics with a Gauduchon connection being "K\"ahler-like", namely, satisfying the same symmetries for curvature as the Levi-Civita and Chern connections. In particular, we investigate $6$-dimensional solvmanifolds with invariant complex structures with trivial canonical bundle and with invariant Hermitian metrics. The results for this case give evidence for two conjectures that are expected to hold in more generality: first, if the Strominger-Bismut connection is K\"ahler-like, then the metric is pluriclosed; second, if another Gauduchon connection, different from Chern or Strominger-Bismut, is K\"ahler-like, then the metric is K\"ahler. As a further motivation, we show that the K\"ahler-like condition for the Levi-Civita connection assures that the Ricci flow preserves the Hermitian condition along analytic solutions.