Uniqueness conjecture for radial decreasing quasi-linear Schrödinger solutions

Let $r_1,r_2>0$ be radial decreasing solutions to

$$-\Delta r_i-r_iDelta r_i^2+\lambda_i r_i=r_i^{p}\qquad\text{in }\mathbb{R}^3,\qquad i=1,2.$$

Assume $\lambda_1,\lambda_2>0$, $\mathcal{E}(r_i)<0$, and

$$\|r_i\|^2_{L^2(\mathbb{R}^3)}=c,\qquad i=1,2.$$

Uniqueness conjecture. Then $\lambda_1=\lambda_2$ and $r_2=r_1(\cdot+\xi)$ for some $xi\in\mathbb{R}^3$.

The conjecture would establish uniqueness, up to translations, of the relevant positive radial solutions and support uniqueness of minimizers for the quasi-linear minimization problem. The source states that uniqueness of minimizers up to translations and multiplication by $\pm1$ is not currently known, although it is conjectured; related uniqueness results are available in the semi-linear case.