  For many systems with second class constraints, the question posed in the
title is answered in the negative. We prove this for a range of systems with
two second class constraints. After looking at two examples, we consider a
fairly general proof. It is shown that, to unravel gauge invariances in second
class constrained systems, it is sufficient to work in the original phase space
itself. Extension of the phase space by introducing new variables or fields is
not required.
