Poincaré Duality of Wonderful Compactifications and Tautological Rings

Let $g \geq 2$. Let $\cal {M}_{g,n}^{rt}$ be the moduli space of $n$-pointed genus $g$ curves with rational tails. Let $\cal {C}_g^n$ be the $n$-fold fibered power of the universal curve over $\cal {M}_g$. We prove that the tautological ring of $\cal {M}_{g,n}^{rt}$ has Poincaré duality if and only if the same holds for the tautological ring of $\cal {C}_g^n$. We also obtain a presentation of the tautological ring of $\cal {M}_{g,n}^{rt}$ as an algebra over the tautological ring of $\cal {C}_g^n$. This proves a conjecture of Tavakol. Our results are valid in the more general setting of wonderful compactifications.