  We carry out a detailed study of the structure of domains of discontinuity
$\Omega_\rho$ in $\mathbb{RP}^3$ of $\text{PSL}_4(\mathbb{R})$-Hitchin
representations $\rho$. We then prove the foliated component $\Omega_\rho^1$ of
$\Omega_\rho$ has exactly two group-invariant foliations by properly embedded
projective line segments and has a unique foliation by properly embedded convex
domains in projective planes. This gives a finiteness counterpart to work of
Guichard and Wienhard. We also prove analogues for the non-foliated component
$\Omega_\rho^2$ and deduce a rigidity of projective equivalences of properly
convex foliated projective structures on unit tangent bundles of surfaces.
