Uniqueness conjecture for exterior quadrilaterals of prescribed angles and conformal modulus

Let $M>0$ be a positive real number. Consider exterior quadrilaterals with prescribed angles and vertices $A_1=1$, $A_2=0$, $A_3$, and $A_4$, and let their conformal modulus be $M$. Uniqueness conjecture. For every $M>0$ there is only one exterior quadrilateral with given angles and vertices $A_1=1$, $A_2=0$, $A_3$, $A_4$, whose conformal modulus equals $M$. The preceding discussion shows that in the general non-convex case there are at most three such quadrilaterals; the conjecture proposes uniqueness, but no resolution is supplied here.