We investigate level-set percolation of the discrete Gaussian free field on
,
, in the strongly percolative regime. We consider the event that the level-set of the Gaussian free field below a level
disconnects the discrete blow-up of a compact set
A from the boundary of an enclosing box. We derive asymptotic large deviation upper bounds on the probability that
... [Show full abstract] the local averages of the Gaussian free field deviate from a specific multiple of the harmonic potential of A, when disconnection occurs. If certain critical levels coincide, which is plausible but open at the moment, these bounds imply that conditionally on disconnection, the Gaussian free field experiences an entropic push-down proportional to the harmonic potential of A. In particular, due to the slow decay of correlations, the disconnection event affects the field on the whole lattice. Furthermore, we provide a certain 'profile' description for the field in the presence of disconnection. We show that while on a macroscopic scale the field is pinned around a level proportional to the harmonic potential of A, it locally retains the structure of a Gaussian free field shifted by a constant value. Our proofs rely crucially on the 'solidification estimates' developed in arXiv:1706.07229 by A.-S. Sznitman and the second author.