Finiteness conjecture for highly connected minimal non-embeddable cell complexes

For an integer $n$, consider $n$-dimensional cell complexes $K$ whose underlying spaces are $(n-1)$-connected and do not embed in $S^{2n}$. An $h$-minor is a cell complex obtained by the $h$-minor operation defined in the source. The finiteness conjecture for minimal non-embeddable cell complexes. For each $n$, there exist only finitely many such cell complexes $K$ for which the underlying space of every proper $h$-minor $L$ embeds in $S^{2n}$.

This predicts a finite obstruction set for embeddings of highly connected $n$-dimensional cell complexes in $S^{2n}$. The supplied text gives no evidence of a resolution.