The dynamics conjecture for the once-punctured Klein bottle Fricke space

Let $\Omega^K$ be the Fricke space of the once-punctured Klein bottle, and let

$$\Omega_c^K=\kappa_{11}^{-1}(c)\cap\Omega^K.$$

Dynamics conjecture. The action of $\Gamma$ on $\Omega^K$ is wandering, and for each $c>2$ its action on

$$\kappa_{11}^{-1}(c)\setminus\Omega_c^K$$

is ergodic.

This describes the proposed decomposition of the Klein-bottle Fricke space into a wandering region and ergodic level-set dynamics. The supplied text does not indicate whether the claim has been proved or disproved.