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. Actually, we show that
Strominger K\"ahler-like is equivalent to the pluriclosedness of the Hermitian
metric plus the parallelness of the torsion, even without the compactness
assumption