Sharp Lieb-Oxford constant for the indirect Coulomb energy

For every $N$-particle probability density on $R^{3N}$ with one-body density $\rho$, $\left\langle \sum_{i<j} e^2/\mid x_i - x_j \mid \right\rangle - (e^2/2) \int \int \rho(x) \rho(y)/\mid x - y \mid\, dx\, dy \ge -C_{\mathrm{LO}} e^2 \int \rho^{4/3}\, dx$. Determine the optimal $C_{\mathrm{LO}}$, known to satisfy $1.4442 \le C_{\mathrm{LO}} \le 1.58$, and decide whether it equals the uniform-electron-gas constant, the low-density limit of the jellium energy per particle in units $e^2 \rho^{1/3}$. That constant is $\ge 1.4442$, the bcc Wigner-crystal value, with equality if mathph.crystallization.jellium-bcc holds.

A proof of the inequality with the conjectured constant, or a trial state whose ratio exceeds the uniform-electron-gas value.