  We study left-invariant conformal Killing $2$- or $3$-forms on simply
connected $2$-step nilpotent Riemannian Lie groups. We show that if the center
of the group is of dimension greater than or equal to 4, then every such form
is automatically coclosed (i.e. it is a Killing form). In addition, we prove
that the only Riemannian 2-step nilpotent Lie groups with center of dimension
at most 3 and admitting left-invariant non-coclosed conformal Killing $2$- and
$3$-forms are: the Heisenberg Lie groups and their trivial 1-dimensional
extensions, endowed with any left-invariant metric, and the simply connected
Lie group corresponding to the free 2-step nilpotent Lie algebra on 3
generators, with a particular 1-parameter family of metrics. The explicit
description of the space of conformal Killing $2$- and $3$-forms is provided in
each case.
