Uniqueness conjecture for the no-spin Hartree–Fock minimisation problem

Let $w$ be a positive radial non-increasing function. For density $\rho\geq 0$ and temperature $T\geq 0$, let $E_{\textnormal{no-spin}}^{\rm HF}(\rho,T)$ denote the no-spin Hartree–Fock minimisation problem. Uniqueness for the no-spin problem. The minimisation problem $E_{\textnormal{no-spin}}^{\rm HF}(\rho,T)$ admits a unique minimiser for every $\rho,T\geq 0$. This conjecture would rule out non-uniqueness at fixed density in the no-spin model, although non-uniqueness at fixed chemical potential may still occur and is relevant to possible spin-symmetry breaking.