  QSO near-zones are an important probe of the the ionization state of the IGM
at z ~ 6-7, at the end of reionization. We present here high-resolution
cosmological 3D radiative transfer simulations of QSO environments for a wide
range of host halo masses, 10$^{10-12.5}$ M_sun. Our simulated near-zones
reproduce both the overall decrease of observed near-zone sizes at 6 < z < 7
and their scatter. The observable near-zone properties in our simulations
depend only very weakly on the mass of the host halo. The size of the H II
region expanding into the IGM is generally limited by (super-)Lyman Limit
systems loosely associated with (low-mass) dark matter haloes. This leads to a
strong dependence of near-zone size on direction and drives the large observed
scatter. In the simulation centred on our most massive host halo, many
sightlines show strong red damping wings even for initial volume averaged
neutral hydrogen fractions as low as ~ 10$^{-3}$. For QSO lifetimes long enough
to allow growth of the central supermassive black hole while optically bright,
we can reproduce the observed near-zone of ULAS J1120+0641 only with an IGM
that is initially neutral. Our results suggest that larger samples of z > 7
QSOs will provide important constraints on the evolution of the neutral
hydrogen fraction and thus on how late reionization ends.
