Holomorphic split submersion property of L and S on the disk model

Determine whether, for any p ≥ 1, the pre-Schwarzian derivative map L: Mp(D*) -> Bp(D) and the Schwarzian derivative map S: Mp(D*) -> Ap(D) are holomorphic split submersions onto their respective images; specifically, ascertain the existence of local holomorphic right inverses and the split-surjectivity of their differentials at every point in their images.

Background

In Section 4 the authors analyze the unit disk model and show that the canonical map J: L(Mp(D*)) -> S(Mp(D*)) is a holomorphic split submersion (Theorem 4.5), and further describe L(Mp(D*)) as a real-analytic disk bundle over S(Mp(D*)) (Theorem 4.7). They also construct global real-analytic sections for p > 1 (Corollary 4.8).

However, beyond the map J between the image spaces, it remains uncertain whether the original maps from Beltrami coefficients on D*, namely the pre-Schwarzian map L: Mp(D*) -> Bp(D) and the Schwarzian map S: Mp(D*) -> Ap(D), themselves enjoy the split submersion property onto their images. This entails the existence of local holomorphic right inverses and complemented kernels of the differentials, which is complicated by the non-injectivity phenomena and normalization constraints discussed in Section 4.