Commutativity of Cho and normal Jacobi operators on real hypersurfaces in the complex quadric

On a real hypersurface in the complex quadric we can consider the Levi-Civita connection and, for any non-zero real constant $k$, the $k$-th generalized Tanaka—Webster connection. We prove the non-existence of real hypersurfaces in the complex quadric for which the covariant derivatives associated to both connections coincide when they act on the normal Jacobi operator of the real hypersurface.