Closed chains of conics carrying poncelet triangles

We investigate closed chains of conics which carry Poncelet triangles. In particular, we show that every chain of conics which carries Poncelet triangles can be closed. Furthermore, for k=3 and k=4 we show that there are closed chains of pairwise conjugate conics which carry Poncelet k-gons such that the contact points of each k-gon are the vertices of the next k-gon—such miraculous chains of conics do not exist for 5≤k≤23