Möbius-rigidity conjecture for homeomorphic critical points of conformal energy

Let $\theta$ be a sufficiently regular homeomorphism $[0,2\pi]\rightarrow[0,2\pi]$ with $\theta(\pi)=\pi$. Suppose that, for every $y\in[0,2\pi]$,

$$\int_0^{2\pi}\left(\cot\left[\frac{\theta(y)-\theta(y-x)}{2}\right]-\cot\left[\frac{\theta(y+x)-\theta(y)}{2}\right]\right)\cos(x)\,dx=0.$$

Möbius-rigidity conjecture. There is an $a>0$ such that

$$\theta(t)=\arctan\left(\frac{(1-a^2)\sin(t)}{(1+a^2)\cos(t)-2a}\right).$$

This asserts that Möbius transformations are the only homeomorphic critical points of the conformal energy functional $\mathcal E:\operatorname{Hom}(\mathbb{S})\rightarrow[1,\infty)$. The supplied text gives no evidence of resolution.