Stress-algebra weak Lefschetz conjecture for homology manifolds

Let $\Delta$ be an orientable simplicial $\mathbb{R}$-homology $d$-manifold without boundary, with $H_1(\Delta;\mathbb{Z}/2\mathbb{Z})=0$ whenever $d\geq2$. Let $\Upsilon$ be a subset of the $\mathbb{Q}$-generic PL realizations of $\Delta$ in $\mathbb{R}^d$. Stress-algebra weak Lefschetz conjecture. There exists a dense subset $\Upsilon$ such that, for every $\nu\in\Upsilon$, the stress algebra $\Psi(\Delta,\nu)$ has the weak Lefschetz property. If true, this would establish Kalai's manifold $g$-conjecture for a large class of orientable $\mathbb{R}$-homology manifolds without boundary.