  We give examples of finite, simplicial $2$-complexes that do not PL embed in
$\mathbb{R}^4$ and exhibit, for each such complex, a family of PL immersions
into $\mathbb{R}^4$ that hide the obstruction to embedding in "higher and
higher order Milnor invariants". In addition, we show that the embedding
obstructions defined by Krushkal in [8] vanish for our examples. We also answer
a question in a paper of Avramidi-Okun-Schreve.
