A divergent horocycle in the horofunction compactification of the Teichm\"uller metric

We give an example of a horocycle in the Teichm"uller space of the five-times-punctured sphere that does not converge in the Gardiner–Masur compactification, or equivalently in the horofunction compactification of the Teichm"uller metric. As an intermediate step, we exhibit a simple closed curve whose extremal length is periodic but not constant along the horocycle. The example lifts to any Teichm"uller space of complex dimension greater than one via covering constructions.