A stronger question, is the deficiency of $G$ realized for a presentation with the minimal number of generators (rank($G$))? This question is asked in a paper of Rapaport, and proved to be true for nilpotent and 1-relator groups.
Addendum:
The question appears as Question 2, p. 2, of a book by Gruenberg. Lubotzky has answered the analogous question affirmatively in the category of profinite groups (Corollary 2.5). (Note though that this is in the category of profinite presentations, so it does not imply an affirmative answer even for finite groups).
