What's the necessary and sufficient condition for gauge equivalence in the projective construction?

Question

The definition of gauge equivalence and notations used here is the same as those in my previous question.

As we know, the condition $\chi_{ij}'=G_i\chi_{ij}G_j^\dagger$(where $G_i\in SU(2)$) is a sufficient condition for the $H$ and $H'$ being gauge equivalent, but is it also a necessary condition? If not, then what's the necessary and sufficient condition for gauge equivalence? Is my conjecture (1) correct?

Thanks in advance.

This post imported from StackExchange Physics at 2014-03-09 08:39 (UCT), posted by SE-user K-boy