Problems in quasi-polynomial time that could conceivably be in P (without causing collapses or violating widely held beliefs)
Group Isomorphism! Although Ricky Demer gave lots (though certainly not all) details on this, there is an important point I want to highlight, esp. given the stated motivation for the question, namely:


  Putting Group Isomorphism into $\mathsf{P}$ is a key obstacle to putting Graph Isomorphism into $\mathsf{P}$


Group Isomorphism (when given by multiplication tables) reduces to Graph Isomorphism, so the above was technically always true. But when Graph Iso was way up at $2^{\tilde{O}(\sqrt{n})}$ it was so far from Group Iso's $2^{(\log n)^2}$ that there were clearly other obstacles in the way. If Graph Iso is in time $2^{(\log n)^{O(1)}}$, Group Isomorphism is then a much more immediately relevant obstacle to putting Graph Isomorphism into $\mathsf{P}$. In particular, this would suggest that Babai's algorithm handles much [1] of the combinatorics of GI, and the problem is now down to hard algebra. (Not that there's not hard algebra in GI, but GroupIso is by definition about algebra.) 

[1] I won't say "all" of the combinatorics, because the exponent in Babai's algorithm is most likely $&gt; 2$, which would still leave a gap between GI and GroupIso. Also because there may still be some combinatorics sitting inside the algebra of GroupIso...