  We demonstrate that all rigidly rotating strings with center of mass at the
origin of the $dS_3$ static patch satisfy the Higuchi bound. This extends the
observation of Noumi et al. for the open GKP-like string to all solutions of
the Larsen-Sanchez class. We argue that strings violating the bound end up
expanding towards the horizon and provide a numerical example. Adding point
masses to the open string only increases the mass/spin ratio. For segmented
strings, we write the conserved quantities, invariant under Gubser's algebraic
evolution equation, in terms of discrete lightcone coordinates describing kink
collisions. Randomly generated strings are found to have a tendency to escape
through the horizon that is mostly determined by their energy. For rapidly
rotating segmented strings with mass/spin $<$ 1, the kink collisions eventually
become causally disconnected. Finally we consider the scenario of cosmic
strings captured by a black hole in dS and find that horizon friction can make
the strings longer.
