Conjecture on dense colorings for typical norms

For a norm $\|\cdot\|$ on $\mathbb{R}^n$, write $\chi(\mathbb{R}^n;\|\cdot\|)$ for the chromatic number of its unit-distance graph, and let $\chi_{ed}(\mathbb{R}^n;\|\cdot\|)$ denote the least number of color classes in a proper coloring whose classes are dense in $\mathbb{R}^n$. A norm is called typical if it belongs to the Baire-category typical set in the space of norms.

Typical-norm dense-coloring conjecture. For a typical norm,

$$\chi_{ed}\bigl(\mathbb{R}^n;\|\cdot\|\bigr)=\chi\bigl(\mathbb{R}^n;\|\cdot\|\bigr)=2^n.$$

For typical norms, the equality $\chi(\mathbb{R}^n;\|\cdot\|)=2^n$ is known, but the coloring establishing it is nonconstructive. The conjecture asserts that requiring every color class to be dense does not increase the chromatic number.