A Coulomb ground-state density without Kohn-Sham ensemble representation

We construct a finite three-electron Coulomb molecule whose spin-summed ground-state density cannot be reproduced by any ground-state ensemble of noninteracting electrons in a single real, spin-independent local potential in $L^{3/2}({\mathbf{R}}^3)+L^{\infty }({\mathbf{R}}^3)$. The molecule has two equal positive integer nuclear charges, specified by an exact finite nonnumerical formula.