  We present results from a deep (200 ks) Chandra observation of the early-type
galaxy NGC 4552 (M89) which is falling into the Virgo cluster. Previous
shallower X-ray observations of this galaxy showed a remnant gas core, a tail
to the South of the galaxy, and twin `horns' attached to the northern edge of
the gas core [machacek05a]. In our deeper data, we detect a diffuse, low
surface brightness extension to the previously known tail, and measure the
temperature structure within the tail. We combine the deep Chandra data with
archival XMM-Newton observations to put a strong upper limit on the diffuse
emission of the tail out to a large distance (10$\times$the radius of the
remnant core) from the galaxy center. In our two previous papers
[roediger15a,roediger15b], we presented the results of hydrodynamical
simulations of ram pressure stripping specifically for M89 falling into the
Virgo cluster and investigated the effect of ICM viscosity. In this paper, we
compare our deep data with our specifically tailored simulations and conclude
that the observed morphology of the stripped tail in NGC 4552 is most similar
to the inviscid models. We conclude that, to the extent the transport processes
can be simply modeled as a hydrodynamic viscosity, the ICM viscosity is
negligible. More generally, any micro-scale description of the transport
processes in the high-$\beta$ plasma of the cluster ICM must be consistent with
the efficient mixing observed in the stripped tail on macroscopic scales.
