Non-log-exp conjecture for non-integrable SR Martinet spheres

Let $\beta$ be the parameter appearing in the non-integrable SR Martinet problem, and let an SR Martinet sphere denote the corresponding sub-Riemannian sphere. A function is pfaffian if it belongs to the Pfaffian class, and the log-exp category is the class of functions used in the integrable case. Non-log-exp conjecture. If $\beta eq 0$, then SR Martinet spheres are not log-exp, and are not even pfaffian. This predicts that the log-exp category is too narrow to describe the non-integrable case; the preceding discussion gives evidence from the loss of analyticity in the pendulum representation, but no general resolution is stated.