Uniquely among the dwarf spheroidal (dSph) satellite galaxies of the Milky Way, Fornax hosts globular clusters. It remains a puzzle as to why dynamical friction has not yet dragged any of Fornax's five globular clusters to the centre, and also why there is no evidence that any similar star cluster has been in the past (for Fornax or any other tidally undisrupted dSph). We set up a suite of 2800 N-body simulations that sample the full range of globular cluster orbits and mass models consistent with all existing observational constraints for Fornax. In agreement with previous work, we find that if Fornax has a large dark matter core, then its globular clusters remain close to their currently observed locations for long times. Furthermore, we find previously unreported behaviour for clusters that start inside the core region. These are pushed out of the core and gain orbital energy, a process we call ‘dynamical buoyancy’. Thus, a cored mass distribution in Fornax will naturally lead to a shell-like globular cluster distribution near the core radius, independent of the initial conditions. By contrast, cold dark matter-type cusped mass distributions lead to the rapid infall of at least one cluster within Δt = 1–2 Gyr, except when picking unlikely initial conditions for the cluster orbits (∼2 per cent probability), and almost all clusters within Δt = 10 Gyr. Alternatively, if Fornax has only a weakly cusped mass distribution, then dynamical friction is much reduced. While over Δt = 10 Gyr this still leads to the infall of one to four clusters from their present orbits, the infall of any cluster within Δt = 1–2 Gyr is much less likely (with probability 0–70 per cent, depending on Δt and the strength of the cusp). Such a solution to the timing problem requires (in addition to a shallow dark matter cusp) that in the past the globular clusters were somewhat further from Fornax than today; they most likely did not form within Fornax, but were accreted.