Conjecture on one-dimensional Möbius presentations of Cha–Nguyen–Tauber systems

The Cha–Nguyen–Tauber procedure associates a solution generating system to a homogeneous quadratic equation over $\mathbb{R}^3$. A 1-D piecewise Möbius transformation is a one-dimensional piecewise transformation whose branches are Möbius transformations. A measure is $\sigma$-finite, absolutely continuous and invariant when it has these properties with respect to the transformation; a dynamical system is conservative and ergodic in the usual measure-theoretic sense. The Cha–Nguyen–Tauber conjecture. Whenever the Cha–Nguyen–Tauber procedure produces a solution generating system, one may construct a presentation of the solution generating system that is a 1-D piecewise Möbius transformation. Additionally, this map possesses a $\sigma$-finite, absolutely continuous and invariant measure, which makes the system conservative and ergodic. The conjecture would establish that every solution generating system produced by the procedure admits a one-dimensional dynamical presentation with strong invariant-measure properties. The paper states that no claim is currently made about whether the construction works in generality, so the conjecture remains open.