  In the present paper, we construct a cusped hyperbolic $4$-manifold with all
cusp sections homeomorphic to the Hantzsche-Wendt manifold, which is a rational
homology sphere. By a result of Gol\'enia and Moroianu, the Laplacian on
$2$-forms on such a manifold has purely discrete spectrum. This shows that one
of the main results of Mazzeo and Phillips from 1990 cannot hold without
additional assumptions on the homology of the cusps. This also answers a
question by Gol\'enia and Moroianu from 2012. We also correct and refine the
incomplete classification of compact orientable flat $3$-manifolds arising from
cube colourings provided earlier by the last two authors.
