The quasi-reductivity characterization of contact Lie poset algebras

Let $\mathfrak{g}$ be an index-one Lie poset algebra of type A, B, C, or D. A Lie algebra is quasi-reductive if it admits a one-form of reductive type, meaning a one-form whose coadjoint stabilizer modulo the center has center consisting of semisimple elements.

Quasi-reductivity conjecture. An index-one Lie poset algebra of type A, B, C, or D is contact if and only if it is quasi-reductive.

The forward implication is motivated by the contact forms constructed for the classical families associated with height-one posets, which are of reductive type, while the converse follows from an argument analogous to the cited type-A result. A full characterization for arbitrary heights remains open.