Uniqueness conjecture for the GJMS equation on the sphere

Let $n\geq 2$, let $\mathbb{S}^n$ be the round sphere, let $P_{\mathbb{S}^n}$ denote its GJMS operator, let $Q_{g_{\mathbb{S}^n}}=(n-1)!/2$ be its constant $Q$-curvature, and let $v_{\alpha}$ be a solution of the corresponding equation. Uniqueness conjecture. For every $\alpha\in[1/2,1)$, the equation

$$\alpha P_{\mathbb{S}^n}v_{\alpha}+2Q_{g_{\mathbb{S}^n}}=2Q_{g_{\mathbb{S}^n}}e^{nv_{\alpha}}$$

has only the trivial solution. This strengthens the question of whether the infimum of the functional $J_{\alpha}$ is zero for $\alpha\in[1/2,1)$, in the setting of high-order conformally invariant equations on the sphere; the supplied text gives no resolution of the conjecture.