Lichnerowicz's conjecture on canonical metrics and conformal isometries

Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n$ that is not conformally diffeomorphic to $(S_n,g_{can})$. Denote by $I(M,\tilde g)$ the isometry group of a metric $\tilde g$ and by $C(M,g)$ the conformal group of $g$.

Lichnerowicz's conjecture. There exists a metric $\tilde g$ conformal to $g$, with constant scalar curvature $R_{\tilde g}$, such that

$$I(M,\tilde g)=C(M,g).$$

The source states that this follows from the equivariant Yamabe problem and from the compactness of the conformal group away from the round sphere; it is therefore presented here as solved.