The manifold algebraic g-conjecture

Let $\Delta$ be a homology manifold, and let $L^i_s(\Delta)$ denote the set of pairs consisting of a linear system of parameters and a one-form satisfying the partial Lefschetz condition in degree $i$ as defined in the source. Manifold algebraic g-conjecture. For every $i\ge\lceil d/2\rceil$,

$$L^i_s(\Delta)\ne\emptyset.$$

This is proposed as an extension of the algebraic g-conjecture from homology spheres to homology manifolds, with or without boundary. Its status is open.