$ASP$-complete reductions, introduced by Ueda and Nagao, relate the hardness of computational problems in $FNP$. Basically, $ASP$-reduction is a polynomial time reduction between instances and a polynomial time computable bijection on solution sets. $ASP$-completeness implies the $NP$-completeness of the corresponding decision problem.
I came up with the following conjecture: There is an $ASP$-reduction between any pair of (natural) $NP$-complete problems.
In other words, every Karp reduction between $NP$-complete problems can be modified by providing polynomial-time computable bijection on solution sets.
Is this a known conjecture? Is there any counterexample? What are the complexity-theoretic consequences? Does it have any implication on the Isomorphism Conjecture of Berman and Hartmanis?
UPDATE For this post, natural problems are the NP-complete problems listed in Garey and Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness (to address Emil's comment). Also, I accept other more general notions of natural NP-complete problems surveyed by Allender. Specifically, NP-complete problems that are either p-isomorphic to SAT or NP-creative or have universal relation.
P.S. Goldreich states that "all known reductions among natural $NP$-complete problems are either parsimonious or can be easily modified to be so". The above conjecture is strengthening of Goldreich's observation. ( Computational Complexity: A Conceptual Perspective By Oded Goldreich, page 204).
N. Ueda and T. Nagao. NP-completeness results for NONOGRAM via parsimonious reductions. Technical Report TR96-0008, Department of Computer Science, Tokyo Institute of Technology, 1996.