Superiority conjecture for the cross-polytope bound

Let $n\ge 2$. The functions $\underline{\alpha}_{\Delta^n}^*$ and $\underline{\alpha}_{\Delta^n}$ satisfy the comparison

Superiority conjecture.

$$\underline{\alpha}_{\Delta^n}^*(\beta)\ge \underline{\alpha}_{\Delta^n}(\beta)$$

for any $\beta\ge 2$, with equality if and only if $\beta=2$ or $\alpha_{\Delta^n}^*(\beta)=\beta$.

This conjecture asserts that the starred family gives at least as strong a bound as the unstarred family; the source reports numerical evidence but no proof.