Consider Theorem $11$ of this paper, which says:

Any depth $d$ circuit that accepts all $n$ bit strings of Hamming weight $\frac{n}{2} + 1$ and rejects all strings of Hamming weight $\frac{n}{2}$ has size $\text{exp}[\Omega(n^{1/(d-1)})].$

Is there a matching upper bound to the given lower bound for the same problem?

Consider Theorem 11 of this paper (S. Aaronson, BQP and the Polynomial Hierarchy), which says:

Any depth $d$ circuit that accepts all $n$ bit strings of Hamming weight $\frac{n}{2} + 1$ and rejects all strings of Hamming weight $\frac{n}{2}$ has size $\exp[\Omega(n^{1/(d-1)})]$.

Is there a matching upper bound to the given lower bound for the same problem?