Metastable Local Disjunction conjecture

Let $D=D^{s_1}\sqcup\ldots\sqcup D^{s_r}$ be a disjoint union of $r$ disks, let $f:D\to B^d$ be a proper PL map such that $f\partial D^{s_1}\cap\ldots\cap f\partial D^{s_r}=\emptyset$, and suppose that

$$rd\ge (s_1+s_2+\ldots+s_r)+s_i+3$$

and $d\ge s_i+3$ for each $i$. If the map

$$f^r:\partial(D^{s_1}\times\ldots\times D^{s_r})\to (B^d)^r-\{(x,x,\ldots,x)\in(B^d)^r\ |\ x\in B^d\}$$

extends to a continuous map of $D^{s_1}\times\ldots\times D^{s_r}$, Metastable Local Disjunction conjecture. then there is a PL map $\overline f:D\to B^d$ such that

$$\overline f=f \quad\text{on}\quad D^{s_r}\cup\partial D\quad\text{and}\quad \overline fD^{s_1}\cap\ldots\cap \overline fD^{s_r}=\emptyset.$$

The claim is presented as one of the paper's conjectures, whose complete proof is the subject of the note; the supplied text does not establish whether it has been resolved.