  An old conjecture in non-K\"ahler geometry states that, if a compact
Hermitian manifold has constant holomorphic sectional curvature, then the
metric must be K\"ahler (when the constant is non-zero) or Chern flat (when the
constant is zero). It is known to be true in complex dimension $2$ by the work
of Balas and Gauduchon in 1985 (when the constant is negative or zero) and
Apostolov, Davidov and Muskarov in 1996 (when the constant is positive). In
dimension $3$ or higher, the conjecture is only known in some special cases,
such as the locally conformally K\"ahler case (when the constant is negative or
zero) by the work of Chen, Chen and Nie, or for complex nilmanifolds with
nilpotent $J$ by the work of Li and the second named author.
  In this note, we confirm the above conjecture for all non-balanced Bismut
torsion parallel (BTP) manifolds. Here the BTP condition means that the Bismut
connection has parallel torsion. In particular, the conjecture is valid for all
Vaisman manifolds.
