Density of conditioned directional lattices near the projective triangle

Let $L$ and $S$ be as in Corollary~, and let

$$\Delta:=\bigl\{p\in\operatorname{Gr}_2(\mathbb{R}^3):p\text{ contains one of the axes }\mathbb{R}e_i,\ i=1,2,3\bigr\}.$$

Let $\mathfrak{D}_S(L)$ denote the corresponding set of conditioned directional lattices, and let $\pi:X\to\operatorname{Gr}_2(\mathbb{R}^3)$ be the natural projection. Projective-triangle conjecture. The closure of $\mathfrak{D}_S(L)$ in $X$ contains $\pi^{-1}(\Delta)$. This is proposed in connection with the expectation that real cubic numbers are well approximable and generic for the Gauss map; the source gives no resolution.