Is every knot isotopic to the unknot?

The following problem was stated by D. Rolfsen in his 1974 paper; according to R. Daverman it was being discussed since the mid-60s. Is every knot in $S^3$ isotopic (=homotopic through embeddings) to a PL knot — or, equivalently, to the unknot? In particular, is the Bing sling isotopic to a PL knot?

We show that the Bing sling $B$ is not isotopic to any PL knot by an isotopy which extends to an isotopy of any 2-component link obtained from $B$ by adding a disjoint component $Q$ such that $lk(B,Q)=1$. Moreover, the assertion remains true if the additional component is allowed to self-intersect, and even to get replaced by a new one at any time instant $t$, as long as it remains disjoint from the original
component $K_t$ and represents the same conjugacy class as the old one in $G/[G’,G”]$, where $G=\pi_1(S^3\setminus K_t)$. The are examples
showing that the latter result cannot be improved in certain ways.

I plan to present a sketch of the proof, modulo some ingredients. The details can be found in arXiv:2406.09365 and the main ingredients in arXiv:2406.09331 and arXiv:math/0312007v3.