A note on almost abelian groups with constant holomorphic sectional curvature

A long-standing conjecture in non-K\"ahler geometry states that if the Chern (or Levi-Civita) holomorphic sectional curvature of a compact Hermitian manifold is a constant c, then the metric must be K\"ahler when c 0 and must be Chern (or Levi-Civita) flat when c=0. The conjecture is known to be true in dimension 2 by the work of Balas-Gauduchon, Sato-Sekigawa, and Apostolov-Davidov-Muskarov in the 1980s and 1990s. In dimension 3 or higher, the conjecture is still open except in some special cases, such as for all twistor spaces by Davidov-Grantcharov-Muskarov, for locally conformally K\"ahler manifolds (when c 0) by Chen-Chen-Nie, etc. In this short note, we consider compact quotients G/ where G is a Lie group equipped with a left-invariant complex structure and a compatible left-invariant metric, and is a discrete subgroup. We confirm the conjecture when the Lie algebra g of G either is almost abelian, or contains a J-invariant abelian ideal of codimension 2.