Density of directional lattices over isotropic planes

Let $Q(v_1,v_2,v_3):=2v_1v_3-v_2^2$ and

$$V_Q^1:=\{v:Q(v)=1\}.$$

Let $\mathcal{C}\subset\operatorname{Gr}_2(\mathbb{R}^3)$ be the circle of isotropic subspaces, and let $\pi:X\to\operatorname{Gr}_2(\mathbb{R}^3)$ be the natural projection. Isotropic-circle conjecture. The closure of $\mathfrak{D}_{V_Q^1}(\mathbb{Z}^3)$ contains

$$\pi^{-1}(\mathcal{C}).$$

The source presents this as an analogue of the preceding conjecture and as a conjecture implying the motivating conjecture; no resolution is supplied.