TITLE: What conjectures in anabelian geometry are false?
QUESTION [14 upvotes]: 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.

REPLY [10 votes]: Mochizuki proved that the anabelian conjecture for hyperbolic curves (the hom-form) only needs the maximal pro-$p$ quotient of the fundamental groups, for some prime $p$, which gives a stronger result than originally envisioned by Grothendieck.
Y. Hoshi (a student of Mochizuki) proved that an analogous generalization of the section conjecture is false:
Existence of nongeometric pro-p Galois sections of hyperbolic curves,
  Publications of the Research Institute for Mathematical Sciences (Publ. Res. Inst. Math. Sci.) 46 (2010), no. 4, 829-848.
You can get this paper from his homepage.