The strong NE-equivalence conjecture for simple homotopy types

Let $\{X_i\}_{i=1}^\infty$ be a family of finite simplicial complexes. The strong NE-equivalence conjecture. There exists an infinite family of finite simplicial complexes $\{X_i\}_{i=1}^\infty$, which all have the same simple homotopy type, such that

$$X_i\not\simeq_{\text{\tt NE}} X_j$$

for all $i\neq j$. This strengthens the weak conjecture by requiring infinitely many pairwise non-NE-equivalent complexes within a single simple homotopy type; its status is left open in the supplied text.