The NE-equivalence conjecture for simple homotopy types

Let $X$ and $Y$ be finite simplicial complexes. The NE-equivalence conjecture. There exist finite simplicial complexes $X$ and $Y$ having the same simple homotopy type, such that $X\not\simeq_{\text{\tt NE}} Y$. The conjecture proposes that NE-equivalence is strictly finer than Whitehead simple homotopy equivalence; the paper presents this as the weak conjecture, with the stronger claim asserting an infinite family of pairwise non-NE-equivalent complexes of one simple homotopy type.