Piercing Spheres and the General Schoenflies Theorem

An embedded $(n-1)$-sphere $S$ in $\mathbb R^n$ is flat if it can be taken by a self homeomorphism of $\mathbb R^n$ to the standard unit sphere. Unfortunately even in lower dimensions like $n=3$ requiring simple connectivity of the complementary domains is not enough to characterise flatness (Fox–Artin). The following question was posed:
Is a $2$-sphere $S$ in $\mathbb R^3$ flat if for any point $x$ of $S$ there is a straight line segment (according to the affine structure of $\mathbb R^3$) intersecting $S$ only in $x$ and having endpoints in separate complemenatry domains?
Such a segment, if it exists, is called piercing. The answer was found out to be negative by Bing. However if one adds certain continuity conditions on the piercing segments, for example, that they form a bicollar around the sphere, then the answer changes and becomes a special case of the general Schoenflies Theorem (Mazur, Brown).
In fact it is possible to say a bit more for $n>3$. That is:
If $S$ is an $(n-1)$-sphere in $\mathbb R^n$, $n>3$, then the set $W$ of points where it fails to be locally flat contains no isolated point. In particular $W$ is either empty or contains a Cantor set.
This remarkable result was built up through the independent methods of Cantrell, Chernavsky and Kirby.

Zoom link: https://mi-ras-ru.zoom.us/j/95004507525
Access code: the Euler characteristic of the wedge of two circles
(the password is not the specified phrase but the number that it determines)