The famous Isomorphism Conjecture of Berman and Hartmanis says that all $NP$-complete languages are polynomial time isomorphic (p-isomorphic) to each other. The key significance of the conjecture is that it implies $P\neq NP$. It was published in 1977, and a piece of supporting evidence was that all $NP$-complete problems known at the time were indeed p-isomorphic. In fact, they were all paddable, which is a nice, natural property, and implies p-isomorphism in a nontrivial way.
Since then, the trust in the conjecture deteriorated, because candidate $NP$-complete languages have been discovered that are not likely to be p-isomorphic to $SAT$, although the problem is still open. As far as I know, however, none of these candidates represent natural problems; they are constructed via diagonalization for the purpose of disproving the Isomorphism Conjecture.
Is it still true, after nearly four decades, that all known natural $NP$-complete problems are p-isomorphic to $SAT$? Or, is there any conjectured natural candidate to the contrary?