The smooth framed-isotopy faithfulness conjecture for tame tangles

A smooth $m$-tame tangle is a tame smooth embedding $f:W\to\mathbb{I}^n$ of an $m$-manifold. A smoothly framed isotopic pair consists of tangles related by a framed isotopy that is smooth as a tame tangle.

Smooth framed isotopy of tame smooth tangles implies diffeomorphism. Given smooth $m$-tame tangles $f:W\to\mathbb{I}^n$ and $f':W'\to\mathbb{I}^n$, if $f$ and $f'$ are smoothly framed isotopic, then $W$ and $W'$ are diffeomorphic.

The PL analogue is stated to be true. The source also suggests that, in the stable range $n>2m+1$, the converse should hold, but that additional expectation is not part of this conjecture's assertion.