Some complete $ω$-powers of a one-counter language in the difference hierarchy

Fix a finite alphabet $Σ$ having at least two letters, and natural numbers $n,k\geq 1$. We denote by $D_n(\mathbfΠ^0_k)$ the class of differences of $n$ subsets of the Cantor space $Σ^ω$ in $\mathbfΠ^0_k$, and by $\checked{D}_n(\mathbfΠ^0_k)$ the class of complements of sets in $D_n(\mathbfΠ^0_k)$.