The positivity conjecture for asymptotic realisation constants

For each integer $d>1$, let $\alpha_d$ denote the corresponding asymptotic growth constant for $d$-realisation numbers. Positivity conjecture. For every $d>1$ we have

$$\alpha_d>1.$$

The conjecture asserts that the asymptotic constant is nontrivial in every dimension greater than one; the source presents it after numerical bounds for $\alpha_2$.