Kalai's strong Lefschetz conjecture for homology manifolds

Let $\Delta$ be a $(d-1)$-dimensional orientable homology manifold over $k$, let $k[\Delta]$ be its face ring, and let $\Theta\subseteq k[\Delta]$ be a linear system of parameters. Set

$$I=\bigoplus_{i=1}^{d-1}\operatorname{Soc}(k[\Delta]/\Theta)_i,$$

and define $\overline{k[\Delta]}=(k[\Delta]/\Theta)/I$. Let $\omega\in k[\Delta]$ be a linear form. Kalai's conjecture. There are $\Theta$ and $\omega$ such that

$$\dim_k\overline{k[\Delta]}_i={\mathfrak h}”_i(\Delta)$$

for $1\leq i\leq\lfloor d/2\rfloor$, and such that multiplication by $\omega^{d-2i}$ induces an isomorphism

$$\omega^{d-2i}:\overline{k[\Delta]}_i\longrightarrow\overline{k[\Delta]}_{d-i}$$

for $1\leq i\leq\lfloor d/2\rfloor$. Part (i) is known from the results cited in the source, while the Lefschetz isomorphism assertion remains the conjectural part and would imply the ${\mathfrak g}$-conjecture for triangulations of spheres.