  We discuss the constant $\sigma_{2}$ problem for conic 4-spheres. Based on
earlier works of Chang-Han-Yang and Han-Li-Teixeira, we are able to find a
necessary condition for the existence problem. In particular, when the
condition is sharp, we have the uniqueness result similar to that of Troyanov
in dimension 2. It indicates that the boundary of the moduli of all conic
4-spheres with constant $\sigma_{2}$ metrics consists of conic spheres with 2
conic points and rotational symmetry.
