In the last days I came to consider the following question which I'd be happy to see answered by the affirmative:

if $f:X\to S$ is a morphism of schemes which is formally étale, universally bijective, and universally schematically dominant, is $f$ an isomorphism?

Comments :

1. you may assume $X$ and $S$ affine if you like,
2. I can show that it is enough to show that $f$ is a monomorphism, i.e. the diagonal morphism $\Delta:X\to X\times_SX$ is an isomorphism. Under the assumptions, $\Delta$ is a closed bijective immersion defined by an ideal $I$ such that $I^2=I$. If $I$ were nilpotent this would imply $I=0$, but in general...