Twisted Patterson-Sullivan densities on nilpotent covers for Anosov subgroups

Let $\mathrm{\Gamma }<\mathsf{S}\mathsf{L}(d,\mathbb{R})$ be a Zariski-dense Borel-Anosov subgroup and let $\phi$ be positive on its limit cone. For every $\chi \in \mathrm{H}\mathrm{o}\mathrm{m}(\mathrm{\Gamma },\mathbb{R})$, we construct twisted $(\phi ,\chi )$-conformal densities and prove their uniqueness, atomlessness, and ergodicity. For every normal subgroup ${\mathrm{\Gamma }}_0⊲\mathrm{\Gamma }$ with nilpotent quotient, we classify the ergodic $\phi$-conformal densities of ${\mathrm{\Gamma }}_0$ in terms of characters of $\mathrm{\Gamma }\mathrm{/}{\mathrm{\Gamma }}_0$.