  We investigate percolation of the vacant set of random interlacements on
$\mathbb{Z}^d$, $d\geq 3$, in the strongly percolative regime. We consider the
event that the interlacement set at level $u$ disconnects the discrete blow-up
of a compact set $A\subseteq \mathbb{R}^d$ from the boundary of an enclosing
box. We derive asymptotic large deviation upper bounds on the probability that
the local averages of the occupation times deviate from a specific function
depending on 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 occupation-time
profile undergoes an entropic push governed by a specific function depending on
$A$. Similar entropic repulsion phenomena conditioned on disconnection by
level-sets of the discrete Gaussian free field on $\mathbb{Z}^d$, $d \geq 3$,
have been obtained by the authors in arxiv:1808.09947. Our proofs rely
crucially on the `solidification estimates' developed in arXiv:1706.07229 by
A.-S. Sznitman and the second author.
