Proving things suspected to be true in anabelian geometry is usually very hard. Maybe it is easier to disprove things suspected to be false?
In particular, I am interested in false generalizations of the main conjecture to higher dimensions.
Is there a pair of non-isomorphic varieties, $X$ and $Y$, which one might naively expect to satisfy an anabelian conjecture, but which in fact have the same etale fundamental group?
Obviously I am not interested in simply-connected varieties, abelian varieties, fibrations with the same fundamental group as their base, and other obvious counterexamples. Thus an example must certainly be of dimension $>1$.
Another question is:
Are there guesses about how exactly to reconstruct the geometry of a variety from its etale fundamental group, that might seem true, but are in fact false?
I am thinking about ideas like the Section Conjecture.