For Siegel modular forms of parallel weight there is a good theory of Fourier-Jacobi expansions (as for example in Faltings-Chai) and I suspect that there is a similar theory for the general case, but I am not able to find a good reference.

So my question is: do we have a Fourier-Jacobi expansion theory for Siegel modular forms of not necessarily parallel weight? If this is true, where can I find a good explanation?

This question is related to my other question here, indeed in the parallel weight case the answer to the other question is yes (because of the projection formula) and I think that a good way to prove it in general is to use Fourier-Jacobi expansion (as Skinner and Urban apparently do, without giving a reference).