Can $\log^k n$ alternations be simulated in $\mathsf{NC}^k$?
Of course the equalities and inclusions claimed in the question hold only for uniform $\mathsf{NC}^k$. The class $\mathsf{AALTSP}(\log^k n, \log n)$ is the same as uniform $\mathsf{AC}^k$, so the question is the same as whether $\mathsf{NC}^k = \mathsf{AC}^k$.

In particular the case $k=1$ implies that $\mathsf{L}$ is equal to $\mathsf{NL}$ and even $\mathsf{LogCFL}$, since $\mathsf{NC}^1 \subseteq \mathsf{L} \subseteq \mathsf{NL} \subseteq \mathsf{LogCFL} \subseteq \mathsf{AC}^1$.