  In this paper, we prove a conjecture raised by Angella, Otal, Ugarte, and
Villacampa recently, which states that if the Strominger connection (also known
as Bismut connection) of a compact Hermitian manifold is K\"ahler-like, in the
sense that its curvature tensor obeys all the symmetries of the curvature of a
K\"ahler manifold, then the metric must be pluriclosed. What we actually showed
is a bit more: for any given Hermitian manifold, the Strominger K\"ahler-like
condition is equivalent to the pluriclosedness of the metric plus the
parallelness of the torsion.
