Metastable Mabillard–Wagner conjecture

Let $s,d,r\ge2$ be integers satisfying

$$rd\ge(r+1)s+3$$

and let $K$ be a finite $s$-dimensional simplicial complex. An almost $r$-embedding is a map $f:K\to\mathbb R^d$ such that $f(\sigma_1)\cap\ldots\cap f(\sigma_r)=\emptyset$ whenever $\sigma_1,\ldots,\sigma_r$ are pairwise disjoint simplices of $K$. Let $\Sigma_r$ be the permutation group on $r$ elements, acting on the sphere $S^{d(r-1)-1}_{\Sigma_r}$ of real $d\times r$ matrices whose row sums are zero and whose squared matrix entries sum to $1$, by permuting columns. Metastable Mabillard–Wagner conjecture. There exists an almost $r$-embedding $f:K\to\mathbb R^d$ if and only if there exists a continuous $\Sigma_r$-equivariant map

$$\bigcup\{\sigma_1\times\cdots\times\sigma_r:\ \sigma_i\text{ is a simplex of }K,\ \sigma_i\cap\sigma_j=\emptyset\text{ for every }i\ne j\}\to S^{d(r-1)-1}_{\Sigma_r}.$$

The claim is stated as a conjecture in the paper and gives an equivariant obstruction-theoretic characterization of almost $r$-embeddability in the metastable range; the supplied text does not provide resolution evidence.