Strong ionization conjecture for atomic Hamiltonians

Let $H_N$ be the $N$-electron atomic Hamiltonian with nuclear charge $Z\in\mathbb{N}$. Strong ionization conjecture. There is a universal constant $C>0$ such that, whenever $N>Z+C$, the operator $H_N$ has no eigenvalue. This would strengthen the ionization conjecture by ruling out all eigenstates beyond a uniformly bounded excess charge; the statement remains open.