Streets-Tian Conjecture on Lie algebras with codimension $2$ abelian ideals

A Hermitian-symplectic metric is a Hermitian metric whose K\"ahler form is given by the $(1,1)$-part of a closed $2$-form. Streets-Tian Conjecture states that a compact complex manifold admitting a Hermitian-symplectic metric must be K\"ahlerian (i.e., admitting a K\"ahler metric). The conjecture is known to be true in dimension $2$ but is open in dimensions $3$ or higher in general, except in a number of special situations, such as twistor spaces (Verbitsky), Fujiki ${\mathcal C}$ spaces (Chiose), Vaisman manifolds (Angella-Otiman), etc. For Lie-complex manifolds (namely, compact quotients $G/\Gamma$ of Lie groups by discrete subgroups with left-invariant complex structures), the conjecture has also been confirmed in a number of special cases, including when $G$ is nilpotent (Enrietti-Fino-Vezzoni), when $G$ is completely solvable (Fino-Kasuya), or when $J$ is abelian (Fino-Kasuya-Vezzoni), or $G$ is almost abelian (Fino-Kasuya-Vezzoni, Fino-Paradiso), etc. In this article, we conduct a detailed case analysis and confirm Streets-Tian Conjecture for $G$ whose Lie algebra contains an abelian ideal of codimension $2$. Such Lie algebras are always solvable of step at most $3$, but are not $2$-step solvable and not completely solvable in general. Our approach is explicit in nature by describing both the Hermitian-symplectic metrics on such Lie algebras and the pathways of deforming them into K\"ahler ones, in hope of advancing our understanding of the subtlety and intricacy of this interesting conjecture in non-K\"ahler geometry.