| Faber and Pandharipande formulated a ``trinity'' of conjectures regarding the tautological rings of moduli spaces of curves. Specifically, they conjectured that there is Poincar\'e duality in the tautological ring of the space of n-pointed genus g curves that are either (i) stable, or (ii) of compact type, or (iii) with rational tails. I will explain that there are now two known counterexamples to this conjecture: in the stable case, it fails when g=2 and n >= 20 (this is due to joint work with Orsola Tommasi), and in the compact type case, it fails when g=2 and n >= 8. |