Sheehan conjectured that there are no 4-regular graphs that are uniquely hamiltonian (i.e. have exactly one hamilton cycle).

Evidence that might be loosely seen to be in favour of this conjecture is: that even 3-regular graphs cannot be uniquely hamiltonian, that regular graphs of high enough valency cannot be uniquely hamiltonian, and that it seems (empirically) difficult to make uniquely hamiltonian graphs that do not have most of their vertices of degree 3.

However all of this seems rather inconsequential compared to Fleischner's examples of uniquely hamiltonian graphs of minimum degree 4, which are biregular with degrees 4 and 14. This suggests that provided you've got enough vertices and enough edges, you can connect up cycle-forcing and cycle-avoiding "gadgets" in various complicated ways to ultimately enforce unique hamiltonicity.

At the moment, I'm leaning towards the position that the conjecture is most likely false, but that the smallest counterexample will have hundreds or thousands of vertices.

I'd be interested in hearing other heuristic arguments in either direction, maybe the poster's own informed opinion (see comments) or one that they have read in the literature, or one that they have heard in seminars etc.