Continuity of the Levy-Lieb and Lieb functionals

Let $\rho,\rho_n\in L^1(\mathbb{R}^d,\mathbb{R}_+)$ satisfy $\sqrt{\rho},\sqrt{\rho_n}\in H^1(\mathbb{R}^d)$. The functionals $F^{(0)}$ and $F^{(0)}_{\mathrm{mix}}$ denote the pure-state and mixed-state Levy–Lieb functionals, respectively.

Continuity conjecture. If

$$\sqrt{\rho_n}\to\sqrt{\rho}\quad\text{in }H^1(\mathbb{R}^d),$$

then

$$F^{(0)}(\rho_n)\to F^{(0)}(\rho)\quad\text{and}\quad F^{(0)}_{\mathrm{mix}}(\rho_n)\to F^{(0)}_{\mathrm{mix}}(\rho).$$

The conjecture would imply that pure-state $v$-representable ground densities are not dense when $d\geq 3$, using densities where the mixed-state functional is strictly smaller than the pure-state functional. The paper gives no proof or disproof.