The PL-manifold structure conjecture for tame tangle manifolds

A tame $m$-tangle is an open or compact tame embedding of an $m$-manifold, and its canonical framed PL structure restricts to a weak PL structure on each tangle manifold.

Tame tangle manifolds are PL manifolds. The canonical PL structures of open or compact tame $m$-tangles restrict to PL $m$-manifold structures on tame tangle manifolds.

The preceding observation only gives weak PL manifold structures, so the conjecture asks whether every link is PL homeomorphic to the standard sphere. The source explicitly describes this as depending on a fundamental open problem.