  We provide a theoretical analysis by means of the nonperturbative functional
renormalization group (NP-FRG) of the corrections to scaling in the critical
behavior of the random-field Ising model (RFIM) near the dimension
$d_{DR}\approx 5.1$ that separates a region where the renormalized theory at
the fixed point is supersymmetric and critical scaling satisfies the $d\to d-2$
dimensional reduction property ($d>d_{DR}$) from a region where both
supersymmetry and dimensional reduction break down at criticality ($d<d_{DR}$).
We show that the NP-FRG results are in very good agreement with recent
large-scale lattice simulations of the RFIM in $d=5$ and we detail the
consequences for the leading correction-to-scaling exponent of the peculiar
boundary-layer mechanism by which the dimensional-reduction fixed point
disappears and the dimensional-reduction-broken fixed point emerges in
$d_{DR}$.
