  In a paper by Angella, Otal, Ugarte, and Villacampa, the authors conjectured
that on a compact Hermitian manifold, if a Gauduchon connection other than
Chern or Strominger is K\"ahler-like, then the Hermitian metric must be
K\"ahler. They also conjectured that if two Gauduchon connections are both
K\"ahler-like, then the metric must be K\"ahler. In this paper, we discuss some
partial answers to the first conjecture, and give a proof to the second
conjecture. In the process, we discovered an interesting `duality' phenomenon
amongst Gauduchon connections, which seems to be intimately tied to the
question, though we do not know if there is any underlying reason for that from
physics.
