Uniqueness conjecture for fixed-mass energy minimizers

Let $d\geqslant2$ and $p>q>1$, and let $I(\lambda)$ be the fixed-mass variational problem

$$I(\lambda)=\inf\left\{\frac12\int_{\mathbb R^d}|\nabla u|^2\,dx+\frac{1}{p+1}\int_{\mathbb R^d}|u|^{p+1}\,dx-\frac{1}{q+1}\int_{\mathbb R^d}|u|^{q+1}\,dx:\ u\in H^1(\mathbb R^d)\cap L^{p+1}(\mathbb R^d),\ \int_{\mathbb R^d}|u|^2\,dx=\lambda\right\}.$$

Let $\lambda_c$ denote the critical mass appearing in the variational theory. Uniqueness conjecture. $I(\lambda)$ admits a unique minimizer for all $\lambda\geqslant\lambda_c$, respectively for all $\lambda>\lambda_c$ when $q=1+4/d$. This would follow from the one-turning-point conjecture for $M$. The source notes that uniqueness is understood at differentiability points and that the remaining global uniqueness assertion is conjectural.