Proof of the inscribed rhombus conjecture

Question

I ask to verify whether one of the aspects of a proof I have proposed regarding the inscribed rhombus conjecture (i.e. that every jordan curve $J$ inscribes a rhombus) is true. This proof is based on the 3blue1brown proof for the inscribed rectangle conjecture:
https://www.youtube.com/watch?v=AmgkSdhK4K8
So, I begin by defining the function $f(A,B) : J \rightarrow \Bbb{R}^2 \times S^{1}$ as: $$f(A,B) = (x,y,\theta)$$ Where theta is a function that is periodic at intervals of $\frac{\pi}{2}$
We apply the equivalence $0 \sim 2\pi$ onto the standard topology on the interval $[0,2\pi]$ (i.e the interval of $\theta$) to form a quotient topology on $[0,2\pi]$ so that it may vary continuously (i.e there is no "jump" from 0 to $2\pi$). Then, we proceed by stating that, as there exists a continuous mapping $M$ from a mobius loop to the input space (unordered pairs of points), the function maps the mobius loop continuously onto $\Bbb{R}^{2} \times S^{1}$ with the boundary mapped onto the jordan curve $J$.
Here, is it guaranteed that this mapping must have a self intersection (thus completing the proof?)