  We derive asymptotic upper and lower bounds on the large deviation
probability that the level set of the Gaussian free field on $Z^d$, d bigger or
equal to three, below a given level disconnects the discrete blow-up of a
compact set A from the boundary of the discrete blow-up of a box that contains
A, when the level set of the Gaussian free field above this level is in a
strongly percolative regime. These bounds substantially strengthen the results
of arXiv:1412.3960, where A was a box and the convexity of A played an
important role in the proof. We also derive an asymptotic upper bound on the
probability that the average of the Gaussian free field well inside the
discrete blow-up of A is above a certain level when disconnection occurs. The
derivation of the upper bounds uses the solidification estimates for porous
interfaces that were derived in the work arXiv:1706.07229 of A.-S. Sznitman and
the author to treat a similar disconnection problem for the vacant set of
random interlacements. If certain critical levels for the Gaussian free field
coincide, an open question at the moment, the asymptotic upper and lower bounds
that we obtain for the disconnection probability match in principal order, and
conditioning on disconnection lowers the average of the Gaussian free field
well inside the discrete blow-up of A, which can be understood as entropic
repulsion.
