Entropy conjecture for tournament digraphs

Let $\mathcal{T}_n$ be the set of $n$-vertex tournaments. For a parameter $\alpha$, let $H_\alpha$ be the corresponding entropy function, let $h_n^\alpha$ be the number of distinct values of $H_\alpha$ over $\mathcal{T}_n$, and let $S_n$ be the number of distinct score sequences of $n$-tournaments.

Entropy conjecture. For $\alpha$ sufficiently large,

$$\lim_{n \to \infty}\frac{h_n^{\alpha}}{S_n} > 1.$$

The authors report this as an observation-based conjecture about how the number of entropy values compares with the number of score sequences as the parameter increases. The supplied text gives no resolution, so the conjecture remains open.