Monotonicity conjecture for the normalized bosonic binding threshold

For each $N\in\mathbb{N}$, let $Z_*(N)$ be the minimum nuclear charge needed to properly bind $N$ bosonic electrons, and define the normalized threshold

$$\Lambda_*(N):=\frac{Z_*(N)}{N}.$$

Monotonicity conjecture. The map $N\mapsto\Lambda_*(N)$ is monotonically increasing.

The source notes that $\Lambda_*(1)=0$, $0<\Lambda_*(2)\leq 1/2$, and that $\Lambda_*(N)$ tends to a limiting constant as $N$ tends to infinity. These facts do not imply the conjectured monotonicity, whose resolution is not given.