Möbius-conjugacy rigidity conjecture for an oval

Let $\gamma$ be an oval. For points $P,Q$, let $f_P,f_Q:\gamma\to\gamma$ be the involutions associated with the points, and set $F_{P,Q}=f_P\circ f_Q$. Consider pairs for which either both points lie in the exterior of the oval and the line $PQ$ intersects the oval, or both points lie in the interior of $\gamma$. Möbius-conjugacy rigidity conjecture. Assume that for all such pairs of points the map $F_{P,Q}:\gamma\to\gamma$ is $C^\infty$-conjugated to a Möbius transformation. Then $\gamma$ is an ellipse. For an ellipse, the relevant maps arise as restrictions of hyperbolic isometries and are Möbius transformations on the circle at infinity; the source presents the converse characterization as a conjecture, with no resolution stated.