Conformality is Equivalence Relation on Set of Riemannian Metrics

Theorem
Let $M$ be a smooth manifold.

Let $\set g$ be the set of Riemannian metrics on $M$.

Suppose $\set g$ is equipped with the conformality relation.

That is, suppose that for any pair of Riemannian metrics there is a conformal transformation.

Then conformality relation is equivalence relation.