Threshold conjecture for proper bosonic atomic ground states

Let $N\in\mathbb{N}$ be the number of bosonic electrons, let $Z>0$ be the nuclear charge, and let $Z_*(N)$ denote the infimum of charges that properly bind $N$ bosonic electrons. Write $\Psi^{(Z,N)}_{\mathrm{Bose}}$ for an $N$-body ground-state wave function of the Coulomb Hamiltonian.

Threshold conjecture. A proper $N$-body ground-state wave function $\Psi^{(Z,N)}_{\mathrm{Bose}}$ exists whenever $Z>Z_*(N)$.

By definition, a ground state at $Z=Z_*(N)$ need not be proper, so the conjecture concerns the entire strict-binding region above the threshold. It is presented as an analogue of a claim discussed by Reed and Simon; the available context does not establish the conjecture's resolution.