The ionization conjecture for atoms

Let $H_N$ be the $N$-electron atomic Hamiltonian with nuclear charge $Z\in\mathbb{N}$, and let $E_N=\inf\sigma(H_N)$. Denote by $N_c=N_c(Z)$ the largest number of electrons for which $E_{N_c}$ has a minimizer. Ionization conjecture. There is a constant $C>0$, possibly $C=1$, such that

$$N_c\leqslant Z+C.$$

The existence of minimizers for $N<Z+1$ is known by Zhislin's theorem, while the uniform bound on the number of excess electrons remains open.