Hebey–Vaugon conjecture on the strict equivariant Yamabe bound

Let $(M^n,g)$ be a smooth compact Riemannian $G$-manifold with $n\geq 3$, and let $[g]^G$ be the conformal class of $G$-invariant metrics. Write $\operatorname{card}(G\cdot p)$ for the cardinality of the orbit of $p\in M$, and let $\sigma(S^n)$ be the Yamabe invariant of the round sphere.

Hebey–Vaugon conjecture. If $(M^n,g)$ is not conformal to the standard metric on $S^n$, or if the action of $G$ has no fixed point, then

$$Y(M,[g]^G)<\sigma(S^n)\inf_{p\in M}\left(\operatorname{card}(G\cdot p)\right)^{\frac{2}{n}}.$$

This strengthens the corresponding non-strict upper bound for equivariant Yamabe constants. It is a conjecture related to the Equivariant Yamabe Problem; the supplied text does not establish a resolution, so its status remains open.