The optimal simplex anticode conjecture in $A_n$

Let $A_n(D)$ be the maximum cardinality of a subset $\mathcal F\subseteq\mathbb Z^n$ whose diameter in the anticode metric $d_{\mathrm a}$ is at most $D$. Let $S_{n,D}$ be the simplex-shaped anticode, and write $P_D=\lfloor D/2\rfloor$ and $Q_D=\lceil D/2\rceil$. Optimal anticode conjecture. For all integers $n,D\ge0$,

$$A_n(D)=M_n(D)\coloneq |S_{n,D}| =\sum_{j=0}^n\binom nj\binom{P_D}{j} \binom{Q_D+n-j}{n-j}.$$

The theorem established in dimension three proves this equality when $n=3$, while the conjecture asks for the corresponding optimality of the simplex construction in every dimension.