  A long-standing conjecture in complex geometry says that a compact Hermitian
manifold with constant holomorphic sectional curvature must be K\"ahler when
the constant is non-zero and must be Chern flat when the constant is zero. The
conjecture is known in complex dimension $2$ by the work of Balas-Gauduchon in
1985 (when the constant is zero or negative) and by Apostolov-Davidov-Muskarov
in 1996 (when the constant is positive). For higher dimensions, the conjecture
is still largely unknown. In this article, we restrict ourselves to pluriclosed
manifolds, and confirm the conjecture for the special case of Strominger
K\"ahler-like manifolds, namely, for Hermitian manifolds whose Strominger
connection (also known as Bismut connection) obeys all the K\"ahler symmetries.
