Improved Hebey-Vaugon conjecture on equivariant Yamabe invariants in dimension 3

Consider a closed connected $3$-manifold $M$ acted diffeomorphically on by a compact Lie group $G$ with at least one orbit of finite cardinality. We show an upper bound for the $G$-equivariant Yamabe invariant ${\sigma }_G(M)$ under certain topological assumptions, which improved a conjecture of Hebey-Vaugon.