The Gorenstein quotient conjecture for homology manifolds

Let $\Delta$ be a simplicial complex homeomorphic to a $(d-1)$-dimensional $\mathbf{k}$-homology manifold, and let

$$\overline{\mathbf{k}[\Delta]}=\bigl(\mathbf{k}[\Delta]/\Theta\bigr)/I$$

be the quotient defined in the surrounding discussion, where $I$ is the sum of the relevant graded socle components.

Gorenstein quotient conjecture. The ring $\overline{\mathbf{k}[\Delta]}$ should be Gorenstein. This is presented as a closely related, potentially weaker conjecture to the preceding socle characterization; its resolution status is not specified in the source.