(Abridged) Cold fronts in cluster cool cores should be erased on short timescales by thermal conduction, unless protected by magnetic fields that are draped parallel to the front surfaces, suppressing conduction perpendicular to the fronts. We present MHD simulations of cold front formation in the core of a galaxy cluster with anisotropic thermal conduction, exploring a parameter space of conduction strengths parallel and perpendicular to the field lines. Including conduction has a strong effect on the temperature of the core and the cold fronts. Though magnetic field lines are draping parallel to the front surfaces, the temperature jumps across the fronts are nevertheless reduced. The field geometry is such that the cold gas below the front surfaces can be connected to hotter regions outside via field lines along directions perpendicular to the plane of the sloshing motions and along sections of the front which are not perfectly draped. This results in the heating of this gas below the front on a timescale of a Gyr, but the sharpness of the density and temperature jumps may still be preserved. By modifying the density distribution below the front, conduction may indirectly aid in suppressing Kelvin-Helmholtz instabilities. If conduction along the field lines is unsuppressed, we find that the characteristic sharp jumps in X-ray emission seen in observations of clusters do not form. This suggests that the presence of sharp cold fronts in hot clusters could be used to place upper limits on conduction in the {it bulk} of the ICM. Finally, the combination of sloshing and anisotropic thermal conduction can result in a larger flux of heat to the core than either process in isolation. While still not sufficient to prevent a cooling catastrophe in the very central ($r sim$ 5 kpc) regions of the cool core, it reduces significantly the mass of cool gas that accumulates outside those radii.
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