Nonexistence of mirror-image isotopies for sufficiently large polygon spaces

Let $Pol_3(n)$ be the space of vectors $(\vec{e}_1,\dots,\vec{e}_n)\in(\mathbb{R}^3)^n$ satisfying

$$\sum_{i=1}^n\vec{e}_i=0,\qquad \sum_{i=1}^n\lvert\vec{e}_i\rvert=2.$$

Thus $Pol_3(n)$ parametrizes closed $n$-gons in $\mathbb{R}^3$ with total length $2$. Let the isotopy in Theorem 1 refer to a one-parameter isometric isotopy of $Pol_3(n)$ to itself joining each closed polygon to its mirror image.

Nonexistence conjecture. For an arbitrary $n$ with $n\geq n_0$, where $n_0$ is sufficiently large, the isotopy of Theorem 1 is not valid, even without the additional requirements that it preserve the lengths of projections onto the two coordinate planes and keep the projection onto the $x$-axis.

The theorem establishes such an isotopy for $n=7$, while the conjecture predicts that this phenomenon fails for all sufficiently large numbers of edges. The claim is motivated by partial and weaker results related to the Kervaire Invariant One Problem.