Nonvanishing of minimizers of the approximate Levy-Lieb functional

Let $F^{\boldsymbol{\alpha},(0)}(r)$ be the approximate Levy–Lieb functional for a discretized density $r$, and let $\psi$ be any minimizer. The zero set is measured by Lebesgue measure in configuration space $\mathbb{R}^{dN}$.

Nonvanishing-minimizer conjecture. Any minimizer $\psi$ of $F^{\boldsymbol{\alpha},(0)}(r)$ satisfies

$$\left|\left\{x\in\mathbb{R}^{dN}:\psi(x)=0\right\}\right|=0.$$

This condition is stated as sufficient for extending the discretized pure excited-state representability theorem from one particle to arbitrary particle number. Its status is not resolved in the paper.