  A well known conjecture in complex geometry states that a compact Hermitian
manifold with constant holomorphic sectional curvature must be K\"ahler if the
constant is non-zero and must be Chern flat if the constant is zero. The
conjecture is confirmed 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 the class of complex nilmanifolds and confirm the
conjecture in that case.
