Equality of the honest-effort and attainable tournament regions

Let $\mathcal{A}_n$ denote the set of score vectors attainable in tournaments on $n$ players, and let $\mathcal{A}^{*}_n$ denote the region generated by the tournament configurations described above. The examples satisfy $\mathcal{A}^{*}_1=\mathcal{A}_1$, $\mathcal{A}^{*}_2=\mathcal{A}_2$, and $\mathcal{A}^{*}_3=\mathcal{A}_3$. Equality conjecture.

$$\mathcal{A}^{*}_n=\mathcal{A}_n,$$

for every $n\in\mathbb{N}$. This would assert that the configurations captured by $\mathcal{A}^{*}_n$ produce exactly all attainable score vectors, despite the map used to construct the region ceasing to be injective for $n\geq 3$.