  The Eridanus II (EriII) 'ultra-faint' dwarf has a large ($15\,\text{pc}$) and
low mass ($4.3\times10^3\,\text{M}_\odot$) star cluster (SC) offset from its
centre by $23\pm3\,\text{pc}$ in projection. Its size and offset are naturally
explained if EriII has a central dark matter core, but such a core may be
challenging to explain in a $\Lambda$CDM cosmology. In this paper, we revisit
the survival and evolution of EriII's SC, focussing for the first time on its
puzzlingly large ellipticity ($0.31^{+0.05}_{-0.06}$). We perform a suite of
960 direct $N$-body simulations of SCs, orbiting within a range of spherical
background potentials fit to ultra-faint dwarf (UFD) galaxy simulations. We
find only two scenarios that come close to explaining EriII's SC. In the first,
EriII has a low density dark matter core (of size $\sim70\,\text{pc}$ and
density $\lesssim2\times10^8\,\text{M}_{\odot}\,\text{kpc}^{-3}$). In this
model, the high ellipticity of EriII's SC is set at birth, with the lack of
tidal forces in the core allowing its ellipticity to remain frozen in for long
times. In the second, EriII's SC orbits in a partial core, with its high
ellipticity owing to its imminent tidal destruction. However, this latter model
struggles to reproduce the large size of EriII's SC, and it predicts
substantial tidal tails around EriII's SC that should have already been seen in
the data. This leads us to favour the cored model. We discuss potential caveats
to these findings, and the implications of the cored model for galaxy formation
and the nature of dark matter.
