  Using high-accuracy variational Monte Carlo based on group-convolutional
neural networks (GCNNs), we obtain the symmetry-resolved low-energy spectrum of
the spin-1/2 Heisenberg model on several highly symmetric fullerene geometries,
including the famous C$_{60}$ buckminsterfullerene. We argue that as the degree
of frustration is lowered in large fullerenes, they display characteristic
features of incipient magnetic ordering: correlation functions show
high-intensity Bragg peaks consistent with N\'eel-like ordering, while the
low-energy spectrum is organised into a tower of states. Competition with
frustration, however, turns the simple N\'eel order into a noncoplanar one.
Remarkably, we find and predict chiral incipient ordering in a large number of
fullerene structures.
