  We consider random interlacements on $ \mathbb{Z}^d$, $d \ge 3$, when their
vacant set is in a strongly percolative regime. Given a large box centered at
the origin, we establish an asymptotic upper bound on the exponential rate of
decay of the probability that the box contains an excessive fraction $\nu$ of
points that are disconnected by random interlacements from the boundary of a
concentric box of double size. As an application we show that when $\nu$ is not
too large, this asymptotic upper bound matches the asymptotic lower bound
derived in a previous work of the author, and the exponential rate of decay is
governed by a certain variational problem in the continuum which involves the
percolation function of the vacant set of random interlacements.
