Given a Peano continuum X, let C(X) be the space of all increasing maximal chains of connected closed subsets of X, and consider the following question: when does the action of the group H(X) of autohomeomorphisms of X on C(X) has a comeager orbit? In other words, when is it possible to say that there is essentially a unique chain on X?

This problem reduces to verifying an instance of the weak-amalgamation property on certain families of finite graphs with some additional structure. By means of this reduction we show that there is no comeager orbit for the aforementioned action if X is the Menger curve or, more generally, if X is any cross-connected Peano continuum. When X is the Menger curve, we prove moreover that the action of H(X) on C(X) is minimal. This implies that, in this case, the universal minimal flow of the group H(X) is not metrizable.