The anisotropy conjecture for homology spheres

Let $\mathbb{F}$ be a field, and let $\Delta$ be an $\mathbb{F}$-homology sphere of dimension $d-1$. Write $\mathbf{k}=\mathbb{F}(a_{i,j})$ for the field of rational functions in variables $a_{i,j}$, let $\mathbf{k}[\Delta]$ be the Stanley–Reisner ring of $\Delta$ over $\mathbf{k}$, and set

$$\mathbf{k}(\Delta)=\mathbf{k}[\Delta]/(\theta_1,\dots,\theta_d),$$

where $\theta_i=\sum_{j=1}^m a_{i,j}x_j$. The sphere $\Delta$ is generically anisotropic over $\mathbb{F}$ if, for every integer $j$ with $1\leq 2j\leq d$ and every nonzero $u\in\mathbf{k}(\Delta)_j$, one has $u^2\neq 0$.

Anisotropy conjecture. For an arbitrary field $\mathbb{F}$, any $\mathbb{F}$-homology sphere is generically anisotropic over $\mathbb{F}$.

This conjecture would extend the theorem of Papadakis and Petrotou, which establishes generic anisotropy for homology spheres over fields of characteristic $2$, and would provide the generic-anisotropy route to the strong Lefschetz property and the $g$-conjecture in arbitrary characteristic.