Density and interior conjectures for shapes of unit lattices

Let $K$subseteq$\mathbb{R}$ be a totally real cubic number field, let $\mathcal{O}$ be an order in $K$, and let $\Delta_{\mathcal{O}^{\times}}$ denote the shape of its unit lattice, viewed as a point of the modular surface $\operatorname{SL}_2(\mathbb{Z})\backslash\mathbb{H}$. Define

$$\Omega=\left\{\Delta_{\mathcal{O}^{\times}}\in\operatorname{SL}_2(\mathbb{Z})\backslash\mathbb{H}:\mathcal{O}\text{ is an order in a totally real cubic number field}\right\}.$$

Shapes of unit lattices conjectures. The closure $\overline{\Omega}$ in the modular surface is non-compact, the closure $\overline{\Omega}$ in the modular surface has non-empty interior, and the set $\Omega$ is dense in the modular surface.

These claims concern the distribution of shapes arising from logarithmic unit lattices of totally real cubic orders. The source presents them as conjectures and does not provide a resolution; it also indicates that explicit curves lie in the closure and outlines a strategy for proving non-empty interior.