Primitive determinant reformulation of the pseudomanifold Hodge–Riemann conjecture

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 $V$. Let $\ell$ be the Lefschetz element, let $\overline{H}^q_{\operatorname{prim}}(\Delta)$ be the primitive part of $\overline{H}^q(\Delta)$, and let $D_{\operatorname{prim},q}$ be the determinant of the induced Hodge–Riemann form on that primitive part. For a subset $F\subseteq V$ of size $d$, write $\operatorname{ord}_{[F]}(D_{\operatorname{prim},q})$ for its order along $[F]$. Primitive determinant conjecture. The element $\ell$ is strong Lefschetz in all degrees, and for every $F\subseteq V$ of size $d$ and every $0<q\leq d/2$,

$$\operatorname{ord}_{[F]}(D_{\operatorname{prim},q})=0.$$

This is stated as an equivalent reformulation of the pseudomanifold extension conjecture, using the orthogonal decomposition into the previous degree and the primitive part. It is therefore not a separate claim, but an equivalent form of the same open conjecture.