  We study the all-to-all connected $XYZ$ (anisotropic-Heisenberg) spin model
with local and collective dissipations, comparing the results of mean field
theory with the solution of the Lindblad quantum evolution. Leveraging the
permutational symmetry of the model [N. Shammah et al., Phys Rev. A 98, 063815
(2018)], we find exactly (up to numerical precision) the steady state up to
$N=95$ spins. We characterize criticality, studying, as a function of the
number of spins $N$, the spin structure factor, the magnetization, the
Liouvillian gap and the Von Neumann entropy of the steady state. Exploiting the
weak $\mathcal{PT}$-symmetry of the model, we efficiently calculate the
Liouvillian gap, introducing the idea of an antigap. For small anisotropy, we
find a paramagnetic-to-ferromagnetic phase transition in agreement with the
mean-field theory. For large anisotropy, instead, we find a significant
discrepancy from the scaling of the low-anisotropy ferromagnetic phase. We also
study other more experimentally-accessible witnesses of the transition, which
can be used for finite-size studies, namely the bimodality coefficient and the
angular averaged susceptibility. In contrast to the bimodality coefficient, the
angular averaged susceptibility fails to capture the onset of the transition,
in striking difference with respect to lower-dimensional studies. We also
analyze the competition between local dissipative processes (which disentangle
the spin system) and collective dissipative ones (generating entanglement). The
nature of the phase transition is almost unaffected by the presence of these
terms. Our results mark a stark difference with the common intuition that an
all-to-all connected system should fall onto the mean-field solution also for
intermediate number of spins.
