Let $\mathsf{ATISP}(f(n), g(n))$ be the class of languages decided by alternating Turing machines that halt in time $f(n)$ using space $g(n)$. Let $\mathsf{AALTSP}(f(n), g(n))$ be the class of languages decided by alternating Turing machines that halt using $f(n)$ alternations and space $g(n)$.
Ruzzo proved that $\mathsf{NC}^k = \mathsf{ATISP}(\log^k n, \log n)$. He also showed that $\mathsf{NC}^k \subseteq \mathsf{AALTSP}(\log^k n, \log n) \subseteq \mathsf{NC}^{k + 1}$.
Is $\mathsf{NC}^k = \mathsf{AALTSP}(\log^k n, \log n)$?