$\mathsf{FP}^{\mathsf{NP}[\log n]}$ versus $\mathsf{FP}^{\mathsf{NP}||}$
This is an open problem, it implies $\mathsf{NP} = \mathsf{RP}$ among other things. 
See the following paper:

Alan Selman. A Taxonomy of Complexity Classes of Functions. Journal of Computer and Systems Sciences 48 (1992), pp. 357-381.

You can get it here:
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.32.7438&amp;rep=rep1&amp;type=pdf