The famous Isomorphism Conjecture of Berman and Hartmanis says that all $NP$-complete languages are polynomial time isomorphic ($p$-isomorphic) to each other. It has been an early attempt (published in 1977) to prove $P\neq NP$. It is still open, and if true, it indeed implies $P\neq NP$.
However, the proof of this implication is essentially trivial: it is proved by referring to the fact that if $P=NP$, then there are finite $NP$-complete languages, which clearly contradicts to the Isomorphism Conjecture, since a finite language cannot be $p$-isomorphic to SAT.
While the above proof is technically correct, I find it somehow disturbing that to prove the implication one needs a finite $NP$-complete language.
Question: Let us consider a weaker version of the conjecture, stating only that all infinite $NP$-complete languages are $p$-isomorphic. Would this still imply $P\neq NP$? Is anything known about this weaker version of the conjecture?