Natural candidate against the Isomorphism Conjecture?
I think the answer is yes, even today there is no known natural problem that is a candidate for violating the Isomorphism Conjecture.

The primary reason is that typically natural NP-complete problems are very easily seen to be paddable, which Berman and Hartmanis showed suffices to be isomorphic to SAT. For natural graph-related problems this typically involves adding extra vertices that are, e.g., disconnected from the graph, or connected in a very particular (but usually obvious) way. For the decision version of optimization problems, it typically involves adding new dummy variables with no constraints on them. And so on.