Uniqueness conjecture for the $v_m$-Yamabe problem

Let $(M^{2m},g)$ be a closed, even-dimensional Riemannian manifold such that the conformal class $\operatorname{Ca}$ is nonempty. If $(M^{2m},g)$ is not conformally equivalent to $(S^{2m},g_0)$, then there exists a unique $u\in C^{\infty}(M)$ such that

$$\int_M u\,dV_g=0$$

and

$$v_m(e^{-2u}g)=\overline{v}.$$

Uniqueness conjecture. Under these hypotheses, the normalized solution of the $v_m$-Yamabe equation is unique. The conjecture is suggested by the variational and geodesic-convexity properties of the functional developed in the paper; the spherical conformal class is excluded because conformal-group symmetries produce nonuniqueness.