Nonexistence of PL isometric embeddings for some PL manifolds

Let $n\geq 3$ and let $M^n$ be an $n$-dimensional $PL$-manifold. A PL isometric embedding is an embedding of $M^n$ into Euclidean space that preserves its piecewise-linear metric. Nonexistence conjecture. For any $n\geq 3$, there exists an $n$-dimensional $PL$-manifold $M^n$ (and in fact, an infinity of such manifolds), that admits no $PL$ isometric embedding in $\mathbb{R}^{n+1}$. This conjecture asserts that the Burago–Zalgaller construction may not provide PL isometric embeddings for all PL manifolds in codimension one, and that infinitely many counterexamples exist in every dimension at least three.