Strict monotonicity conjecture for the mass of the TFDW Euler–Lagrange solution

Let $c_{TF}>0$, and for $0<\mu<\frac{15}{64c_{TF}}$ let $Q_\mu$ be the unique positive solution of the associated Euler–Lagrange equation, with mass $M(\mu)$, and let $J_{\mathbb{R}^3}(\lambda)$ denote the TFDW variational problem at mass $\lambda$. Strict monotonicity conjecture. The function

$$\left(0;\frac{15}{64c_{TF}}\right)\longrightarrow(0;+\infty),\qquad \mu\longmapsto M(\mu)$$

is strictly increasing and one-to-one. Consequently, for any $0<\mu<\frac{15}{64c_{TF}}$, there exists a unique minimizer $Q_\mu$ of $J_{\mathbb{R}^3}(\lambda)$, up to a phase and a space translation. Numerical computations support the strict increase of $M$, while the preceding result establishes only that each mass has finitely many corresponding values of $\mu$; proving strict monotonicity would yield uniqueness of minimizers modulo phase and translation.