Odd-multiplicity determinant conjecture for Hodge–Riemann forms on pseudomanifolds

Let $k$ be a field of arbitrary characteristic, and let $\Delta$ be a connected oriented simplicial pseudomanifold over $k$ of dimension $d-1$ with vertex set $V$. For $0\leq q\leq d/2$, let $D_q$ denote the determinant of the Hodge–Riemann form on $\overline{H}_F^q(\Delta)$, and let $F\subseteq V$ have size $d$. Write $\operatorname{ord}_{[F]}(D_q)$ for the order of vanishing of $D_q$ along $[F]$. Odd-multiplicity determinant conjecture.

$$\operatorname{ord}_{[F]}(D_q)= \begin{cases} 1 & \text{if }F\text{ is a facet of }\Delta,\\ 0 & \text{otherwise}. \end{cases}$$

This is the proposed extension of the determinant formula for the Hodge–Riemann form from homology manifolds to pseudomanifolds in arbitrary characteristic. The supplied text gives no resolution, so the conjecture remains open.