The inscribed square conjecture for conformal angle spaces

Let $(\mathcal X,\mathcal B,\angle)$ be an angle space conformal to a Jordan curve, meaning that it arises from the angle structure induced by a Jordan curve under conformal equivalence. A tetragon is a four-point set $\{A,B,C,D\}\subset\mathcal X$ with its cyclic angle data. Inscribed square conjecture. There exists a tetragon $\{A,B,C,D\}$ such that

$$\angle(A,B,C)=\frac{\pi}{2},\quad \angle(B,C,D)=\frac{\pi}{2},\quad \angle(C,D,A)=\frac{\pi}{2},\quad \angle(D,A,B)=\frac{\pi}{2},$$

and the remaining angles in the tetragon equal $\pi/4$. This is an angle-space formulation of the classical Inscribed Square Problem for Jordan curves, whose general case is an old unresolved problem.