Excess electron number of atoms bounded independently of nuclear charge

For $H_{N,Z} = \sum_{i=1}^N (-\hbar^2/(2m) \Delta_i - Z e^2/|x_i|) + \sum_{i<j} e^2/|x_i - x_j|$ on antisymmetric $L^2((R^3 x \{\mathrm{up},\mathrm{down}\})^N)$ with a static point nucleus, let $E(N,Z)$ be its ground-state energy and $N_c(Z)$ the largest N with $E(N,Z) < E(N-1,Z)$. Prove or disprove $N_c(Z) \le Z + C$ with C independent of Z. Answer: a proof with explicit or existential C, or a sequence $Z_k$ with $N_c(Z_k) - Z_k$ unbounded.

A proof that bounds $N_{c}(Z) - Z$ uniformly in $Z$, for example by controlling the screened nuclear potential felt by the outermost electrons in the many-body ground state.