  For fibred boundary and fibred cusp metrics, Hausel, Hunsicker, and Mazzeo
identified the space of $L^2$ harmonic forms of fixed degree with the images of
maps between intersection cohomology groups of an associated stratified space
obtained by collapsing the fibres of the fibration at infinity onto its base.
In the present paper, we obtain a generalization of this result to situations
where, rather than a fibration at infinity, there is a Riemannian foliation
with compact leaves admitting a resolution by a fibration. If the associated
stratified space (obtained now by collapsing the leaves of the foliation) is a
Witt space and if the metric considered is a foliated cusp metric, then no such
resolution is required.
