Given that ASP-reductions by definition are parsimonious and parsimonious reductions preserve #P-completeness, one might think that the counting version of all ASP-complete problems are also #P-complete (note that e.g. #3SAT is #P-complete and 3SAT is ASP-complete).
However, ASP-completeness implies that all solutions have polynomial length, whereas no such requirement is there for #P.
Is it the case that all ASP-complete problems are #P-complete? Is there some kind of relationship in the other direction as well?