Stress-algebra dimension formula for homology manifolds

Let $(\Delta,\nu)$ be a PL realization of an orientable simplicial $\mathbb{R}$-homology $d$-manifold without boundary in $\mathbb{R}^d$, and let $\Psi(\Delta,\nu)$ be its stress algebra. Stress-algebra dimension conjecture. If the vertices are realized in general position, then

$$\dim_{\mathbb{R}}\bigl(\Psi_{d+1-i}(\Delta,\nu)\bigr)=h^{\prime\prime}_i(\Delta)\quad\text{for all }0\leq i\leq d+1.$$

This is proposed alongside the weak Lefschetz conjecture as an ingredient toward Kalai's manifold $g$-conjecture for a broad class of orientable $\mathbb{R}$-homology manifolds.