The [**Isomorphism Conjecture**][1] of Berman and Hartmanis states that all $NP$-complete sets are polynomial time isomorphic to each other. This means that $NP$-complete problems are efficiently reducible to each other via polynomial time computable and invertible bijections. The conjecture implies $P\neq NP$. 

The isomorphism conjecture implies an exponential lower bound on the density of $NP$-complete sets since Satisfiability problem is dense. I am wondering if it also implies an exponential lower bound on the density of witnesses for $NP$-complete set.

> Does the isomorphism conjecture imply exponential lower bounds on witnesses density? Does it imply that $NP$-complete problems can not be in [$FewP$][2]?

Density $D$ of a set $S$ refers to the number of strings of length less than $n$ in the language. A set $S$ is exponentially dense if its density is $D=\Omega(2^{n^\epsilon})$ for some $\epsilon \gt 0$ and for infinitely many $n$ and sparse if $D$= $O(poly(n))$.

[1]: http://en.wikipedia.org/wiki/Berman%E2%80%93Hartmanis_conjecture
[2]: https://complexityzoo.uwaterloo.ca/Complexity_Zoo:F#fewp