Eliminating higher-multiplicity intersections in the metastable dimension range

The $r$-fold analogues of Whitney trick were ‘in the air’ since 1960s. However, only in 2010s they were stated, proved and applied to obtain interesting results, most notably by Mabillard and Wagner. Here we prove and apply a version of the $r$-fold Whitney trick when general position $r$-tuple intersections have positive dimension.
Theorem. {\it Assume that $D=D_1\sqcup\ldots\sqcup D_r$ is disjoint union of $n$-dimensional disks, $f:D\to B^d$ a proper PL (smooth) map such that $f\partial D_1\cap\ldots\cap f\partial D_r=\emptyset$ and $rd\ge (r+1)n+3$. If the map
$$f^r:\partial(D_1\times\ldots\times D_r)\to (B^d)^r-\{(x,x,\ldots,x)\in(B^d)^r\ |\ x\in B^d\}$$
extends to $D_1\times\ldots\times D_r$, then there is a PL (smooth) map $\overline f:D\to B^d$ such that
$$\overline f=f \quad\text{on}\quad D_r\cup\partial D\quad\text{and}\quad \overline fD_1\cap\ldots\cap \overline fD_r=\emptyset.$$
} range