Let $r \ge 2$, let $K$ be a finite simplicial complex of dimension $k$, and let $d$ satisfy \begin{align*} rd \ge (r+1)k+3. \end{align*} Write \begin{align*} K^{\times r}_{\Delta}=\{(x_1,\dots,x_r)\in K^r:x_i\in \sigma_i,\ \sigma_1,\dots,\sigma_r \text{ pairwise disjoint simplices of }K\}. \end{align*} If there exists an $S_r$-equivariant map \begin{align*} K^{\times r}_{\Delta}\longrightarrow S(W_r^{\oplus d}), \end{align*} then $K$ admits a PL map $g:K\to \mathbb R^d$ such that \begin{align*} g(\sigma_1)\cap\cdots\cap g(\sigma_r)=\varnothing \end{align*} for every $r$ pairwise disjoint simplices $\sigma_1,\dots,\sigma_r$ of $K$.